Method for calculating and optimizing an eyeglass lens taking into consideration higher-order imaging errors

ABSTRACT

Method for calculating or optimizing a spectacle lens, including
         specifying at least one surface for the spectacle lens to be calculated or optimized;   determining the course of a main ray through at least one visual point of the at least one surface;   determining a first primary set and a second primary set of coefficients of the local aberration of a local wavefront;   specifying at least one function which assigns a second secondary set of coefficients to a second primary set of coefficients, said second secondary set of coefficients defining the higher-order aberration of a propagated wavefront;   determining a higher-order aberration of a local wavefront propagated starting from the at least one visual point along the main ray depending on at least the second primary set of coefficients on the basis of the specified function; and   calculating or optimizing the at least one surface of the spectacle lens based on the determined higher-order aberration of the propagated local wavefront.

BACKGROUND

The present invention relates to a method, a device, and a correspondingcomputer program product for calculating (optimizing) and producing aspectacle lens taking into consideration higher-order aberrations ofboth the eye and the spectacle lens.

For the production or optimization of spectacle lenses, in particular ofindividual spectacle lenses, each spectacle lens is manufactured suchthat the best possible correction of a refractive error of therespective eye of the spectacles wearer is obtained for each desireddirection of sight or each desired object point. In general, a spectaclelens is said to be fully corrective for a given direction of sight ifthe values sphere, cylinder, and axis of the wavefront upon passing thevertex sphere match with the values for sphere, cylinder, and axis ofthe prescription for the eye having the visual defect. In the refractiondetermination for an eye of a spectacles wearer, dioptric values(particularly sphere, cylinder, cylinder axis) for a far (usuallyinfinite) distance and optionally (for multifocal or progressive lenses)an addition for a near distance (e.g. according to DIN 58208) aredetermined. In this way, the prescription (in particular sphere,cylinder, cylinder axis, and optionally addition) that is sent to aspectacles manufacturer is stipulated. In modern spectacle lenses,object distances deviating from the standard, which are used in therefraction determination, can be indicated additionally.

However, a full correction for all directions of sight at the same timeis normally not possible. Therefore, the spectacle lenses aremanufactured such that they achieve a good correction of visual defectsof the eye and only small aberrations in the main regions of use,especially in central visual regions, while larger aberrations arepermitted in peripheral regions.

In order to be able to manufacture a spectacle lens in this way, thespectacle lens surfaces or at least one of the spectacle lens surfacesis first of all calculated such that the desired distribution of theunavoidable aberrations is effected thereby. This calculation andoptimization is usually performed by means of an iterative variationmethod by minimization of a target function. As a target function,particularly a function F having the following functional connectionwith the spherical power S, the magnitude of the cylindrical power Z,and the axis of the cylinder a (also referred to as “SZA” combination)is taken into account and minimized:

$F = {\sum\limits_{i = 1}^{m}\;{\left\lbrack {{g_{i,{S\;\Delta}}\left( {S_{\Delta,i} - S_{\Delta,i,{target}}} \right)}^{2} + {g_{i,{Z\;\Delta}}\left( {Z_{\Delta,i} - Z_{\Delta,i,{target}}} \right)}^{2} + \ldots}\mspace{14mu} \right\rbrack.}}$

In the target function F, at the evaluation points i of the spectaclelens, at least the actual refractive deficits of the spherical powerS_(Δ,i) and the cylindrical power Z_(Δ,i) as well as target values forthe refractive deficits of the spherical power S_(Δ,i,target) and thecylindrical power Z_(Δ,i,target) are taken into consideration.

It was found in DE 103 13 275 that it is advantageous to not indicatethe target values as absolute values of the properties to be optimized,but as their deviation from the prescription, i.e. as the requiredmisadjustment. This has the advantage that the target values areindependent of the prescription (Sph_(V),Zyl_(V),Axis_(V),Pr_(V),B_(V))and that the target values do not have to be changed for everyindividual prescription. Thus, as “actual” values of the properties tobe optimized, not absolute values of these optical properties are takeninto account in the target function, but the deviations from theprescription. This has the advantage that the target values can bespecified independent of the prescription and do not have to be changedfor every individual prescription.

The respective refractive deficits at the respective evaluation pointsare preferably taken into consideration with weighting factors g_(i,SΔ)and g_(i,ZΔ). Here, the target values for the refractive deficit of thespherical power S_(Δ,i,target) and/or the cylindrical powerZ_(Δ,i,target), particularly together with the weighting factor g_(i,SΔ)and g_(i,ZΔ), form the so-called spectacle lens design. In addition,particularly further residues, especially further parameters to beoptimized, such as coma and/or spherical aberration and/or prism and/ormagnification and/or anamorphic distortion, etc., can be taken intoconsideration, which is particularly implied by the expression “+ . . .”.

In some cases, this can contribute to a clear improvement particularlyof an individual adjustment of a spectacle lens if in the optimizationof the spectacle lens not only aberrations up to the second order(sphere, magnitude of astigmatism, and cylinder axis), but alsohigher-order aberrations (e.g. coma, trefoil, spherical aberration) aretaken into consideration.

It is known from the prior art to determine the shape of a wavefront foroptical elements and particularly spectacle lenses that are delimited byat least two refractive boundary surfaces. For example, this can be doneby means of a numerical calculation of a sufficient number ofneighboring rays, along with a subsequent fit of the wavefront data byZernike polynomials. Another approach is based on local wavefronttracing in the refraction (cf. WO 2008/089999 A1). Here, only one singleray (the main ray) per visual point is calculated, accompanied by thederivatives of the vertex depth of the wavefront according to thetransversal coordinates (perpendicular to the main ray). Thesederivatives can be formed up to a specific order, wherein the secondderivatives describe the local curvature properties of the wavefront(such as refractive power, astigmatism) and the higher derivatives areconnected with the higher-order aberrations.

In the tracing of light through a spectacle lens, the local derivativesof the wavefront are calculated at a suitable position in the course ofthe ray in order to compare them with desired values obtained from therefraction of the spectacle lens wearer. This position can be the vertexsphere, for example. In this respect, it is assumed that a sphericalwavefront starts at the object point and propagates up to the firstspectacle lens surface. There, the wavefront is refracted andsubsequently propagates to the second spectacle lens surface, where itis refracted again. If further surfaces exist, the alternation ofpropagation and refraction will be continued until the last boundarysurface has been passed. The last propagation takes place from this lastboundary surface to the vertex sphere.

WO 2008/089999 A1 discloses the laws of refraction at refractivesurfaces not only for aberrations or optical properties of second order,but also for higher orders. If a wavefront with local derivatives knownup to a specific order is obliquely incident on a boundary surface, thevertex depth of which can itself be described by known local derivativesup to the same order, then the local derivatives of the outgoingwavefront can be calculated up to the same order with the calculationmethods according to WO 2008/089999 A1. Such a calculation, particularlyup to the second order, is very helpful for assessing the imageformation properties or optical properties of a spectacle lens in thewearing position. Specifically, such a calculation is of greatimportance if a spectacle lens is to be optimized in the wearingposition over all visual points.

Even if the process of refraction can be described and calculated veryefficiently therewith, the consideration of higher-order aberrationsremains very expensive nevertheless, since especially the requirediterative ray tracing for the propagation of the wavefronts involvesgreat computing effort.

SUMMARY

It is the object of the invention to provide an improved method forcalculating or optimizing a spectacle lens, preferably a progressivespectacle lens, wherein the spectacle lens is adapted to the individualneeds of the spectacles wearer in an improved way. This object is solvedby a computer-implemented method, a device, a computer program product,and a storage medium with the features indicated in the independentclaims. Preferred embodiments are subject of the dependent claims.

According to a first aspect, the invention provides acomputer-implemented method for calculating or optimizing a spectaclelens. First of all, the method comprises specifying at least one surfacefor the spectacle lens to be calculated or optimized. This surfaceparticularly serves as a starting surface for the further individualoptimization. Preferably, the surface is the back surface of thespectacle lens. Preferably, a corresponding starting surface isspecified both for the front surface and for the back surface of thespectacle lens. In a preferred embodiment, only one surface isiteratively varied or optimized during the optimization process. Theother surface of the spectacle lens may be a simple spherical orrotationally symmetric aspherical surface. However, it is also possibleto optimize both surfaces.

Starting from the at least one specified surface, the method comprisesdetermining the course of a main ray through at least one visual point(i) of the at least one surface. Particularly preferably, the course ofthe main ray is determined starting from an object point through thespectacle lens up to a reference surface or assessment surface orcomparative surface, in particular the vertex sphere.

At the at least one visual point (i), a first primary set (s_(io)) and asecond primary set (e_(iok)) of coefficients of the local aberration ofa local wavefront going out from the at least one visual point (i) aredetermined in a vicinity of the main ray, wherein the first primary setof coefficients (s_(io)) specifies the spherical and astigmaticaberration of the outgoing local wavefront and the second primary set ofcoefficients (e_(iok)) specifies at least one further higher-orderaberration (k>2) of the outgoing local wavefront. The outgoing wavefront(or later also referred to as the original wavefront) is the result ofthe refraction of light on the at least one surface. If the one surfaceis the front surface, the outgoing wavefront preferably reflects thelocal refraction properties of the surface in addition to the objectdistance (and the thus involved vergence of the wavefront before hittingsaid surface). The process of refraction could be determinedparticularly by analogy with the description in WO 2008/089999 A1 or inJ. Opt. Soc. Am. A 27, 218-237.

Now, the invention suggests specifying a function e_(pk)=ƒ(e_(ok)) whichassigns a second secondary set of coefficients (e_(pk)) to a secondprimary set of coefficients (e_(ok)), said second secondary set ofcoefficients specifying the at least one higher-order aberration of apropagated wavefront. Specifically, by means of this function, it isvery easily possible to determine the propagation of a wavefront on thebasis of coefficients of the original wavefront.

Thus, the method according to the invention comprises determining atleast one higher-order aberration of a local wavefront propagatedstarting from the at least one visual point (i) along the main raydepending on at least the second primary set of coefficients (e_(iok))on the basis of the specified function e_(pk)=ƒ(e_(ok)). Specifically,to this end, a first and a second secondary set of coefficients of thepropagated wavefront are determined. While the spherical and astigmaticcurvature of the wavefront, i.e. the first secondary set ofcoefficients, are determined particularly on the basis of the firstprimary set of coefficients basically in a known manner, the inventionsuggests a very efficient method for determining the second secondaryset of coefficients as well without being dependent on complex raytracing like in conventional methods.

For the benefit of a clear representation of aspects of the invention,the following terminology for the designation of the coefficients willbe used within the scope of this description. Here, the coefficients orsets of coefficients used or determined for describing the outgoing(original) local wavefront will be referred to as “primary” coefficientsor “primary” sets of coefficients. Accordingly, the coefficients or setsof coefficients used or determined for describing the propagated localwavefront will be referred to as “secondary” coefficients or “secondary”sets of coefficients. Moreover, the coefficients or sets of coefficientsused or determined for describing the spherical and astigmaticaberration of the respective local wavefront will be referred to as“first” coefficients or “first” sets of coefficients, while thecoefficients or sets of coefficients used or determined for describingthe aberrations, i.e. higher-order aberrations, deviating from thespherical and astigmatic aberrations will be referred to as “second”coefficients or “second” sets of coefficients.

After the refraction-induced influence of the spectacle lens on thewavefront was very efficiently determined for higher-order aberrations(as described in e.g. WO 2008/089999 A1 or J. Opt. Soc. Am. A 27,218-237), the method according to the invention now comprisescalculating or optimizing the at least one surface of the spectacle lenstaking into consideration the determined aberration or higher-orderaberrations of the propagated local wavefront. Preferably, a comparisonof the influence on the wavefront, caused by the spectacle lens to beoptimized, with the required prescription is performed at the vertexsphere taking higher-order aberrations into consideration. Depending onthe deviation, at least one of the surfaces of the spectacle lens isvaried and the process is repeated correspondingly until the spectaclelens comes sufficiently close to the required power.

In this way, the invention allows a very fast and precise considerationof higher-order aberrations with clearly reduced computing effort. Thus,an improvement of the individual adjustment of a spectacle lens can beachieved without substantially increasing the computing effort.

Preferably, specifying at least one function e_(pk)=ƒ(e_(ok)) comprisesspecifying a linear function e_(pk)=B_(k)(e_(ok)+r_(k)), or the functione_(pk)=ƒ(e_(ok)) is specified as a linear functione_(pk)=B_(k)(e_(ok)+r_(k)) in which the proportionality term B_(k)depends on the first primary set of coefficients (s_(io)), but not onthe second primary set of coefficients (e_(ok)). Preferably, theremainder term r_(k) does also not depend on the second primary set ofcoefficients (e_(ok)). Particularly preferably, it holds that r_(k)=0,i.e. e_(pk)=B_(k)e_(ok). These embodiments result in a very efficientoptimization particularly for the consideration of third-orderaberrations (k=3).

For a particularly efficient consideration of third-order aberrations(e.g. coma), it is preferred that determining a first primary set ofcoefficients comprise determining a power vector

${s_{o} = \begin{pmatrix}S_{oxx} \\S_{axy} \\S_{oyy}\end{pmatrix}},$wherein determining a second primary set of coefficients comprisesdetermining a coma vector

${e_{o\; 3} = \begin{pmatrix}E_{oxxx} \\E_{oxxy} \\E_{oxyy} \\E_{oyyy}\end{pmatrix}},$and wherein the function

$e_{p\; 3} = {{B_{3}e_{o\; 3}} = {{\gamma^{3}\begin{pmatrix}\beta_{y}^{- 3} & {3\beta_{y}^{- 2}\frac{d}{n}S_{xy}} & {3{\beta_{y}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2}} & \left( {\frac{d}{n}S_{xy}} \right)^{3} \\{\beta_{y}^{- 2}\frac{d}{n}S_{xy}} & {\beta_{y}^{- 1}\left( {\frac{1}{\gamma} + {3\left( {\frac{d}{n}S_{xy}} \right)^{2}}} \right)} & {{\frac{2}{\gamma}\frac{d}{n}S_{xy}} + {3\left( {\frac{d}{n}S_{xy}} \right)^{3}}} & {\beta_{x}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2} \\{\beta_{y}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2} & {{\frac{2}{\gamma}\frac{d}{n}S_{xy}} + {3\left( {\frac{d}{n}S_{xy}} \right)^{3}}} & {\beta_{x}^{- 1}\left( {\frac{1}{\gamma} + {3\left( {\frac{d}{n}S_{xy}} \right)^{2}}} \right)} & {\beta_{x}^{- 2}\frac{d}{n}S_{xy}} \\\left( {\frac{d}{n}S_{xy}} \right)^{3} & {3{\beta_{x}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2}} & {3\beta_{x}^{- 2}\frac{d}{n}S_{xy}} & \beta_{y}^{- 3}\end{pmatrix}}e_{o\; 3}}}$${{{with}\mspace{14mu}\gamma} = \frac{1}{\left. {1 - {\frac{d}{n}S_{oxx}} - \left( {\frac{d}{n}S_{oxy}} \right)^{2} - {\frac{d}{n}S_{oyy}} + {\left( \frac{d}{n} \right)^{2}S_{oxx}S_{oyy}}} \right)}},{\beta_{x} = {{\frac{1}{1 - {\frac{d}{n}S_{oxx}}}\mspace{14mu}{and}\mspace{14mu}\beta_{y}} = \frac{1}{1 - {\frac{d}{n}S_{oyy}}}}}$is specified as at least one function e_(p3)=ƒ(e_(o3)).

In a preferred embodiment, determining the second primary set ofcoefficients (e_(iok)) comprises determining at least a plurality ofprimary subsets (e_(iok), for k=3, 4, . . . ) of the second primary setof coefficients (e_(iok)). Here, specifying at least one functione_(pk)=ƒ(e_(ok)) preferably comprises specifying a linear functione_(pk)=B_(k)(e_(ok)+r_(k)) for each subset (i.e. for each k=3, 4, . . .) such that for each of the linear functions (i.e. for each k=3, 4, . .. ) the proportionality term B_(k) depends on the first primary set ofcoefficients (s_(io)), but not on the second primary set of coefficients(e_(ok)). Preferably, the remainder term r_(k) ₀ of a linear functionfor the subset of the order k₀ does also not depend on the coefficientsof the respective subset k₀, but at most on coefficients of thepreceding subset or the preceding subsets (i.e. the coefficients of theorder k<k₀). To this end, the subsets (e_(iok), for k=3, 4, . . . ) ofthe second primary set of coefficients (e_(iok)) are preferablydetermined such that they form a series of consecutive subsets in aspecific or specifiable sequence of ascending orders. In particular,each subset represents aberrations of the corresponding order, whichcorresponds e.g. to the order (k=3, 4, . . . ) of associated Zernikepolynomials of the respective aberrations.

In a particularly preferred embodiment, determining the second primaryset of coefficients (e_(iok)) comprises determining at least one firstprimary subset (e_(io3)) and one second primary subset (e_(io4)) of thesecond primary set of coefficients (e_(iok)). Here, specifying at leastone function e_(pk)=ƒ(e_(ok)) preferably comprises specifying a firstlinear function e_(p3)=B₃(e_(o3)+r₃) in which the proportionality termB₃ depends on the first primary set of coefficients (s_(io)), but not onthe second primary set of coefficients (e_(ok)). Preferably, theremainder term r₃ does also not depend on the second primary set ofcoefficients (e_(ok)). Particularly preferably, it even holds that r₃=0,i.e. e_(p3)=B₃e_(o3). In addition, specifying at least one functione_(pk)=ƒ(e_(ok)) in this embodiment preferably comprises specifying asecond linear function e_(p4)=B₄(e_(o4)+r₄) in which the proportionalityterm B₄ depends on the first primary set of coefficients (s_(io)), butnot on the second primary set of coefficients (e_(ok)). Preferably, theremainder term r₄ does not depend on the second primary subset (e_(io4))of the second primary set of coefficients (e_(ok)). This results in avery efficient optimization particularly for the consideration offourth-order aberrations (k=4). Particularly preferably, the remainderterm r₄ of the second linear function depends on the first primarysubset (e_(io3)) of the second primary set of coefficients (e_(ok)).

For a particularly efficient consideration of fourth-order aberrations(e.g. spherical aberration), it is preferred that determining a firstprimary set of coefficients comprise determining a power vector

${s_{o} = \begin{pmatrix}S_{oxx} \\S_{axy} \\S_{oyy}\end{pmatrix}},$wherein determining a second primary set of coefficients comprisesdetermining a coma vector

$e_{o\; 3} = \begin{pmatrix}E_{oxxx} \\E_{oxxy} \\E_{oxyy} \\E_{oyyy}\end{pmatrix}$and determining a spherical aberration vector

${e_{o\; 4} = \begin{pmatrix}E_{oxxxx} \\E_{oxxxy} \\E_{oxxyy} \\E_{oxyyy} \\E_{oyyyy}\end{pmatrix}},$andwherein the function

$e_{p\; 4} = {\begin{pmatrix}\beta_{x}^{4} & \ldots & \; & \ldots & 0 \\\vdots & {\beta_{x}^{3}\beta_{y}^{1}} & \; & \; & \vdots \\\; & \; & {\beta_{x}^{2}\beta_{y}^{2}} & \; & \; \\\vdots & \; & \; & {\beta_{x}^{1}\beta_{y}^{3}} & \vdots \\0 & \; & \; & \ldots & \beta_{y}^{4}\end{pmatrix}\left( {e_{o\; 4} + {\frac{d}{n}\begin{pmatrix}{3\left( {{\beta_{x}E_{oxxx}^{2}} + {\beta_{y}E_{oxxy}^{2}} - \frac{S_{oxx}^{4}}{n^{2}}} \right)} \\{3\;{E_{oxxy}\left( {{\beta_{x}E_{oxxx}} + {\beta_{y}E_{oxyy}}} \right)}} \\{{\beta_{x}\left( {{2\; E_{oxxy}^{2}} + {E_{oxxx}E_{oxyy}}} \right)} + {\beta_{y}\left( {{2\; E_{oxyy}^{2}} + {E_{oxxy}E_{oyyy}}} \right)} - \left( \frac{S_{oxx}S_{oyy}}{n} \right)^{2}} \\{3\;{E_{oxyy}\left( {{\beta_{x}E_{oxxy}} + {\beta_{y}E_{oyyy}}} \right)}} \\{3\left( {{\beta_{x}E_{oxyy}^{2}} + {\beta_{y}E_{oyyy}^{2}} - \frac{S_{oyy}^{4}}{n^{2}}} \right)}\end{pmatrix}}} \right)}$$\mspace{20mu}{{{with}\mspace{14mu}\beta_{x}} = {{\frac{1}{1 - {\frac{d}{n}S_{oxx}}}\mspace{14mu}{and}\mspace{14mu}\beta_{y}} = \frac{1}{1 - {\frac{d}{n}S_{oyy}}}}}$is specified as at least one function e_(pk)=ƒ(e_(ok)). Here, thecoefficients of the coma vector particularly form the first primarysubset, and the coefficients of the spherical aberration vectorparticularly form the second primary subset of the second primary set ofcoefficients.

Preferably, the method further comprises determining an angle α betweena first plane of refraction of the main ray at a first surface of thespectacle lens and a second plane of refraction of the main ray at asecond surface of the spectacle lens, wherein determining a higher-orderaberration comprises

-   -   determining a second secondary set (e_(ipk)) of coefficients of        the local aberration of the propagated wavefront; and    -   determining a transformed second secondary set ({tilde over        (e)}_(ipk)) of coefficients depending on the determined angle α,        particularly by applying a rotation matrix R_(k)(α). Thus, for        each refraction step, a Cartesian coordinate system can        preferably be selected such that an axis of a Cartesian        coordinate system is perpendicular to the plane of refraction        and remains unchanged upon transition through the refractive        surface, which results in an easy calculation of the refraction        process.

Preferably, the method further comprises collecting prescription orrefraction data V, wherein the prescription data comprises data withrespect to the spherical power Sph_(V), the magnitude of the astigmatismZyl_(V), the astigmatism axis Axis_(V), as well as at least one furtherpredetermined higher-order refraction HOA_(V).

Preferably, the method comprises minimizing a target function in whichvalues of higher-order aberrations are explicitly taken intoconsideration, e.g. in the form of

${\min\; F} = {{\sum\limits_{i}\;{G_{R,i}\left( {{R_{actual}(i)} - {R_{target}(i)}} \right)}^{2}} + {G_{A,i}\left( {{A_{actual}(i)} - {A_{target}(i)}} \right)}^{2} + {G_{C,i}\left( {{C_{actual}(i)} - {C_{target}(i)}} \right)}^{2} + {G_{S,i}\left( {{S_{actual}(i)} - {S_{target}(i)}} \right)}^{2} + \ldots}$

-   R_(actual)(i) actual refractive error (difference spectacle lens and    eye) at the i^(th) evaluation point-   R_(target)(i) required refractive error at the i^(th) evaluation    point-   G_(R,i) weighting of the refractive error at the i^(th) evaluation    point-   A_(actual)(i) actual astigmatic difference (difference spectacle    lens and eye) at the i^(th) evaluation point-   A_(target)(i) required astigmatic difference at the i^(th)    evaluation point-   G_(A,i) weighting of the astigmatic difference at the i^(th)    evaluation point    and additionally-   C_(actual) (i) actual difference of the coma of the spectacle lens    and the eye at the i^(th) evaluation point-   C_(target)(i) required difference of the coma at the i^(th)    evaluation point-   G_(C,i) weighting of the coma at the i^(th) evaluation point-   S_(actual) (i) actual difference of the spherical aberration of the    spectacle lens and the eye at the i^(th) evaluation point-   S_(target)(i) required difference of the spherical aberration at the    i^(th) evaluation point-   G_(S,i) weighting of the spherical aberration at the i^(th)    evaluation point.

If the target function is extended, the degree of overdetermination willincrease, whereby the stability of the optimization method candeteriorate. Therefore, it is preferred that the target function is notextended, i.e. that the higher-order aberrations are not explicitlytaken into account in the target function, but that both thehigher-order aberrations of the eye and of the spectacle lens are takeninto account in the respective spherocylindrical values. In this case,the method preferably comprises minimizing a target function in whichvalues of higher-order aberrations are implicitly taken intoconsideration over transformed values of the refractive error and theastigmatic difference, e.g. in the form of

${\min\; F} = {{\sum\limits_{i}\;{G_{R,i}\left( {{R_{{actual},t}(i)} - {R_{target}(i)}} \right)}^{2}} + {G_{A,i}\left( {{A_{{actual},t}(i)} - {A_{target}(i)}} \right)}^{2}}$

-   R_(actual,t)(i) transformed refractive error (difference spectacle    lens and eye) at the i^(th) evaluation point-   R_(target)(i) required refractive error at the i^(th) evaluation    point-   G_(R,i) weighting of the refractive error at the i^(th) evaluation    point-   A_(actual,t)(i) transformed astigmatic difference (difference    spectacle lens and eye) at the i^(th) evaluation point-   A_(target)(i) required astigmatic difference at the i^(th)    evaluation point

G_(A,i) weighting of the astigmatic difference at the i^(th) evaluationpoint.

In a preferred embodiment, the method comprises determining atransformed first secondary set of coefficients from the first secondaryset of coefficients and the second secondary set of coefficients.Specifically, a transformed first secondary coefficient is determinedfor every first secondary coefficient of the set of first secondarycoefficients in dependence on at least one coefficient of the secondsecondary set of coefficients, i.e. in dependence on at least onehigher-order aberration. The transformed first secondary coefficientsthen specifically define the transformed spherocylindrical refraction.

For specifying the way in which the transformed spherocylindricalrefraction (in minus cylinder notation) depends on the higher-orderaberrations in preferred embodiments, preferably the following functionare provided:

${{sph}\left( {u,v,{w;r_{0}}} \right)} = {\frac{4\sqrt{3}}{r_{0}^{2}}\left( {u + \frac{\sqrt{2}\sqrt{v^{2} + w^{2}}}{2}} \right)}$${{zyl}\left( {v,{w;r_{0}}} \right)} = {{- \frac{4\sqrt{3}}{r_{0}^{2}}}\sqrt{2}\sqrt{v^{2} + w^{2}}}$${a\left( {v,{w;r_{0}}} \right)} = {\frac{1}{2}{\arctan\left( {w,v} \right)}}$with ${\arctan\left( {x,y} \right)}:=\left\{ \begin{matrix}{{\arctan\left( {y/x} \right)},} & {x > 0} \\{{{\arctan\left( {y/x} \right)} + \pi},} & {{x < 0},{y > 0}} \\{\pi,} & {{x < 0},{y = 0}} \\{{{\arctan\left( {y/x} \right)} - \pi},} & {{x < 0},{y < 0}}\end{matrix} \right.$

In an embodiment in which the higher-order aberrations are at least notexplicitly used for the transformation of the spherocylindrical values,a transformed refraction is preferably determined by:Sph(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m) };r ₀)=sph(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ²;r ₀)Zyl(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m) };r ₀)=zyl(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ²;r ₀)A(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m) };r ₀)=sph(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ;r₀),where c₂ ⁰, c₂ ⁻²,c₂ ² represent second-order Zernike coefficients andc_(n) ^(m) with n≧3 higher-order Zernike coefficients of the wavefrontin the notation of the OSA standard, cf. for example Jason Porter et al.“Adaptive Optics for Vision Science”, Wiley (2006), p. 522. Theparameter r₀ represents the pupil radius. To this end, the methodpreferably comprises detecting a pupil radius r₀. Moreover, the methodpreferably comprises determining second-order Zernike coefficients (c₂⁰,c₂ ⁻²,c₂ ²).

A preferred embodiment taking into consideration the higher-orderaberrations in the determination of the transformed spherocylindricalrefraction, i.e. in a transformation of the spherocylindrical values independence on the second secondary coefficients, is determined by:Sph(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m)})=sph(c ₂ ⁰ ,c ₂ ⁻²+√{square rootover ( 5/3)}c ₄ ⁻² ,c ₂ ²+√{square root over ( 5/3)}c ₄ ²)Zyl(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m)})=zyl(c ₂ ⁻²+√{square root over (5/3)}c ₄ ⁻² ,c ₂ ²+√{square root over ( 5/3)}c ₄ ²)A(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m)})=a(c ₂ ⁻²+√{square root over (5/3)}c ₄ ⁻² ,c ₂ ²+√{square root over ( 5/3)}c ₄ ²)

A further alternative preferred embodiment is determined by:Sph(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m)})=sph(c ₂ ⁰+√{square root over (5/3)}c ₄ ⁻² ,c ₂ ²+√{square root over ( 5/3)}c ₄ ² ,c ₂ ²+√{square rootover ( 5/3)}c ₄ ²)Zyl(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m)})=zyl(c ₂ ⁻²+√{square root over (5/3)}c ₄ ⁻² ,c ₂ ²+√{square root over ( 5/3)}c ₄ ²)A(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m)})=a(c ₂ ⁻²+√{square root over (5/3)}c ₄ ⁻² ,c ₂ ²+√{square root over ( 5/3)}c ₄ ²)

A further preferred embodiment is determined by:Sph(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m)})=sph(c ₂ ⁰+2√{square root over(15)}c ₄ ⁰ ,c ₂ ⁻²+3√{square root over (15)}c ₄ ⁻² ,c ₂ ²+3√{square rootover (15)}c ₄ ²)Zyl(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m)})=zyl(c ₂ ⁻²+3√{square root over(15)}c ₄ ⁻² ,c ₂ ²+3√{square root over (15)}c ₄ ²)A(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² ,{c _(n) ^(m)})=a(c ₂ ⁻²+3√{square root over(15)}c ₄ ⁻² ,c ₂ ²+3√{square root over (15)}c ₄ ²)

Slightly more general, a preferred embodiment is determined by:Sph(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² {c _(n) ^(m)})=sph(c ₂ ⁰ +Δc ₂ ⁰ ,c ₂ ⁻² +Δc ₂⁻² ,c ₂ ² Δc ₂ ²)Zyl(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² {c _(n) ^(m)})=zyl(c ₂ ⁻² +Δc ₂ ⁻² ,c ₂ ² Δc ₂²)A(c ₂ ⁰ ,c ₂ ⁻² ,c ₂ ² {c _(n) ^(m)})=a(c ₂ ⁻² +Δc ₂ ⁻² ,c ₂ ² Δc ₂ ²)in which the changes Δc₂ ⁰, Δc₂ ⁻², Δc₂ ² are functions (particularlycontinuous functions) of the higher-order Zernike coefficients c_(n)^(m) of the wavefront, where n≧3.

Preferably, collecting prescription data comprises collecting firstprescription data for a first object distance and second prescriptiondata for a second object distance. Specifically, the influence of theobject distance on the pupil size can be taken into considerationindividually. In this way, a changed pupil size mainly influences thehigher-order aberrations.

Preferably, the method further comprises:

-   -   specifying an object distance model A1(x, y), where A1        designates the object distance and (x,y) a visual point or        visual spot of the spectacle lens in a specified or specifiable        direction of sight;    -   specifying a function r₀=g (A1), which describes the dependence        of a pupil size r₀ on the object distance A1;    -   determining a pupil size for the at least one main ray (10) on        the basis of the object distance model A1(x, y) and the        specified function r₀=g(A1).

Preferably, the spectacle lens to be optimized is a progressivespectacle lens.

In a further aspect, the invention provides a device for calculating andoptimizing a spectacle lens, comprising:

-   -   a surface model database for specifying at least one surface for        the spectacle lens to be calculated or optimized;    -   main ray determining means for determining the course of a main        ray (10) through at least one visual point (i) of said at least        one surface;    -   primary coefficient determining means for determining a first        primary set (s_(io)) and a second primary set (e_(iok)) of        coefficients of the local aberration of a local wavefront (18)        going out from the at least one visual point (i) in a vicinity        of the main ray (10), wherein the first primary set of        coefficients (s_(io)) specifies the spherical and astigmatic        aberration of the outgoing local wavefront and the second        primary set of coefficients (e_(iok)) specifies a further        higher-order aberration (k>2) of the outgoing local wavefront;    -   a propagation model database for specifying at least one        function e_(pk)=ƒ(e_(ok)), which assigns a second secondary set        of coefficients (e_(pk))) to a second primary set of        coefficients (e_(ok)), said second secondary set of coefficients        specifying the at least one higher-order aberration of a        propagated wavefront;    -   secondary coefficient determining means for determining a        higher-order aberration of a local wavefront propagated starting        from the at least one visual point (i) along the main ray        depending on at least the second primary set of coefficients        (e_(iok)) on the basis of the specified function        e_(pk)=ƒ(e_(ok)); and    -   calculating or optimizing means adapted to calculate or optimize        the at least one surface of the spectacle lens taking into        consideration the determined higher-order aberration of the        propagated local wavefront.

Further, the invention provides a computer program product adapted, whenloaded and executed on a computer, to perform a method for calculatingor optimizing a spectacle lens according to the present invention,particularly in a preferred embodiment thereof.

Moreover, the invention provides a storage medium with a computerprogram stored thereon, the computer program being adapted, when loadedand executed on a computer, to perform a method for calculating oroptimizing a spectacle lens according to the present invention,particularly in a preferred embodiment thereof.

In addition, the invention provides a method for producing a spectaclelens, comprising:

-   -   calculating or optimizing a spectacle lens according to the        method for calculating or optimizing a spectacle lens according        to the present invention, particularly in a preferred embodiment        thereof;    -   manufacturing the thus calculated or optimized spectacle lens.

Moreover, the invention provides a device for producing a spectaclelens, comprising:

-   -   calculating and optimizing means adapted to calculate or        optimize a spectacle lens according to the present invention,        particularly in a preferred embodiment thereof;    -   machining means adapted to finish the spectacle lens.

Further, the invention provides a use of a spectacle lens, producedaccording to the preferred production method, in a predetermined averageor individual wearing position of the spectacle lens in front of theeyes of a specific spectacles wearer, for correcting a visual defect ofthe spectacles wearer.

DRAWINGS

Preferred embodiments of the invention will be described by way ofexample in the following with reference to the accompanying drawings,which show:

FIG. 1 a schematic illustration of the physiological and physical modelof a spectacle lens along with a ray course in a predetermined wearingposition;

FIG. 2 a schematic illustration of a coordinate system with an originalwavefront and a propagated wavefront;

FIG. 3 a schematic illustration of a spherical wavefront with a vergencedistance s_(o) at a distance d from a propagated wavefront with thevergence distance s_(p);

FIG. 4 a schematic illustration of the process of propagation of awavefront;

FIG. 5 a schematic illustration of local coordinate systems of arefractive surface, an incoming and an outgoing wavefront; and

FIG. 6 a flow chart for illustrating a method for optimizing anindividual spectacle lens according to an embodiment of the invention.

DETAILED DESCRIPTION

FIG. 1 shows a schematic illustration of the physiological and physicalmodel of a spectacle lens in a predetermined wearing position along withan exemplary ray course, on which an individual spectacle lenscalculation or optimization according to a preferred embodiment of theinvention is based.

Here, preferably only one single ray (the main ray 10) is calculated pervisual point of the spectacle lens, but further also the derivatives ofthe vertex depths of the wavefront according to the transversalcoordinates (perpendicular to the main ray). These derivatives are takeninto consideration up to the desired orders, wherein the secondderivatives describe the local curvature properties of the wavefront andthe higher derivatives are related to the higher-order aberrations.

In the tracing of light through the spectacle lens, the localderivatives of the wavefronts are ultimately determined at a suitableposition in the ray course in order to compare them with the requiredvalues of the refraction of the spectacles wearer there. In a preferredembodiment, this position is for example the vertex sphere or theentrance pupil of the eye 12. To this end, it is assumed that aspherical wavefront originates at an object point and propagates to thefirst spectacle lens surface 14. There, it is refracted and subsequentlyit propagates (ST2) up to the second spectacle lens surface 16, where itis refracted again. If further surfaces to be considered exist, thealternation of propagation and refraction is continued until the lastboundary surface has been passed, and the last propagation (ST4) thentakes place from this last boundary surface to the vertex sphere (or theentrance pupil of the eye).

In the following, the propagation of the wavefront according to apreferred embodiment of the present invention will be described in moredetail. These statements can e.g. be applied to the propagation of thewavefront between the two spectacle lens surfaces and/or to thepropagation of the wavefront from the rear spectacle lens surface to thevertex sphere.

As illustrated in FIG. 2, preferably a Cartesian coordinate system (withan x axis, a y axis, and a z axis) is defined, the origin of which beingat the intersection point of the main ray 10 with the original wavefront18 for a predetermined main ray 10. The z axis preferably points in thedirection of the main ray 10. The directions of the x axis and the yaxis are preferably selected to be perpendicular to the z axis andperpendicular to each other such that the coordinate system isright-handed. If the original wavefront is assumed to be a wavefront ata refractive surface, i.e. a surface of the spectacle lens, the x axisand/or the y axis is preferably selected to be parallel to the surfaceor surface tangent in the penetration point of the main ray. In anotherpreferred embodiment, the x axis and the y axis are selected to beparallel to the main curvatures of the original wavefront 18.

Preferably, a description of the wavefront according tow(x,y)=(x,y,w(x,y))  (1)is assumed, where the value w(x,y) is represented by

$\begin{matrix}{{w\left( {x,y} \right)} = {\sum\limits_{k = 0}^{\infty}\;{\sum\limits_{m = 0}^{k}\;{\frac{a_{m,{k - m}}}{{m!}{\left( {k - m} \right)!}}x^{m}y^{k - m}}}}} & (2)\end{matrix}$by means of the coefficients

$\begin{matrix}{a_{m,{k - m}} = \left. {\frac{\partial^{k}}{{\partial x^{m}}{\partial y^{k - m}}}{w\left( {x,y} \right)}} \middle| {}_{{x = 0},{y = 0}}. \right.} & (3)\end{matrix}$

Thus, the connection between the coefficients a_(k) _(x) _(,k) _(y) andthe local aberrations E_(k) _(x) _(,k) _(y) can be described by:E _(k) _(x) _(,k) _(y) =na _(k) _(x) _(,k) _(y) E _(2,0) =S _(xx) na_(In,2,0) E _(1,1) =S _(xy) =na _(1,1) E _(0,2) =S _(yy) =na _(0,2) E_(3,0) =na _(3,0)

For aberrations up to the second order, the propagation of a sphericalwavefront with the vergence S_(o)=n/s_(o) of the original wavefront in asurrounding around a main ray can preferably be expressed in a knownmanner by the propagation equation

$\begin{matrix}{S_{p} = {\frac{1}{1 - {\frac{d}{n}S_{o}}}S_{o}}} & (4)\end{matrix}$where S_(p)=n/s_(p) designates the vergence of the propagated wavefront.As illustrated in FIG. 3, s_(o) and s_(p) designate the vertex distanceof the original wavefront 18 and the propagated wavefront 20,respectively, (distance along the main ray 10 from the wavefront to theimage point 22). n designates the refractive index and d the propagationdistance.

By an extension to three dimensions, the spherocylindrical form of thewavefront can be represented as follows. First of all, the curvatures1/s_(o) and 1/s_(p) are identified with the second derivatives of thevertex depths of the original wavefront 18 and the propagated wavefront20, respectively. In the three-dimensional representation, the twoderivatives w_(o) ^((2,0))=∂²w_(o)/∂x², w_(o) ^((1,1))=∂²w_(o)/∂x∂y,w_(o) ^((0,2))=∂²w_(o)/∂y² the original wavefront 18 and correspondinglyfor the propagated wavefront 20 are respectively summarized in form of avergence matrix:

$\begin{matrix}{{S_{o} = {\begin{pmatrix}S_{oxx} & S_{oxy} \\S_{oxy} & S_{oyy}\end{pmatrix} = {n\begin{pmatrix}w_{o}^{({2,0})} & w_{o}^{({1,1})} \\w_{o}^{({1,1})} & w_{o}^{({0,2})}\end{pmatrix}}}},{S_{p} = {\begin{pmatrix}S_{pxx} & S_{pxy} \\S_{pxy} & S_{pyy}\end{pmatrix} = {n\begin{pmatrix}w_{p}^{({2,0})} & w_{p}^{({1,1})} \\w_{p}^{({1,1})} & w_{p}^{({0,2})}\end{pmatrix}}}}} & (5)\end{matrix}$According to

$\begin{matrix}{{S_{oxx} = {\left( {{Sph} + \frac{Cyl}{2}} \right) - {\frac{Cyl}{2}\cos\; 2\alpha}}}{S_{oxy} = {{- \frac{Cyl}{2}}\sin\; 2\alpha}}{S_{oyy} = {\left( {{Sph} + \frac{Cyl}{2}} \right) + {\frac{Cyl}{2}\cos\; 2\alpha}}}} & (6)\end{matrix}$(and analogously for the propagated wavefront) the components of therespective vergence matrix are connected with the known parameters ofspherical power Sph, the magnitude Cyl of cylindrical power, and thecylinder axis a of the cylindrical power. By means of the representationin form of the vergence matrix, by analogy with equation (4), thepropagation of an astigmatic wavefront can be represented via thegeneralized propagation equation

$\begin{matrix}{S_{p} = {\frac{1}{1 - {\frac{d}{n}S_{o}}}S_{o}}} & (7)\end{matrix}$with the identity matrix

$1 = {\begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix}.}$Equivalent to this representation in form of the vergence matrix,

$\begin{matrix}{{s_{o} = {\begin{pmatrix}S_{oxx} \\S_{oxy} \\S_{oyy}\end{pmatrix} = {n\begin{pmatrix}w_{o}^{({2,0})} \\w_{o}^{({1,1})} \\w_{o}^{({0,2})}\end{pmatrix}}}},{s_{p} = {\begin{pmatrix}S_{pxx} \\S_{pxy} \\S_{pyy}\end{pmatrix} = {n\begin{pmatrix}w_{p}^{({2,0})} \\w_{p}^{({1,1})} \\w_{p}^{({0,2})}\end{pmatrix}}}}} & (8)\end{matrix}$are introduced as power vectors in the three-dimensional vector spacefor the original wavefront 18 and the propagated wavefront 20.

Now, for consideration of higher-order aberrations in the propagation ofthe wavefront, corresponding vectors e_(k) of the dimension k+1 areintroduced:

$\begin{matrix}{{e_{ok} = {\begin{pmatrix}E_{{ox}\mspace{14mu}\ldots\mspace{14mu}{xx}} \\E_{{ox}\mspace{14mu}\ldots\mspace{14mu}{xy}} \\\vdots \\E_{{oy}\mspace{14mu}\ldots\mspace{14mu}{yy}}\end{pmatrix}:={n\begin{pmatrix}w_{o}^{({k,0})} \\w_{o}^{({{k - 1},1})} \\\vdots \\w_{o}^{({0,k})}\end{pmatrix}}}},{e_{pk} = {\begin{pmatrix}E_{{px}\mspace{14mu}\ldots\mspace{14mu}{xx}} \\E_{{px}\mspace{14mu}\ldots\mspace{14mu}{xy}} \\\vdots \\E_{{py}\mspace{14mu}\ldots\mspace{14mu}{yy}}\end{pmatrix}:={n\begin{pmatrix}w_{o}^{({k,0})} \\w_{o}^{({{k - 1},1})} \\\vdots \\w_{o}^{({0,k})}\end{pmatrix}}}}} & (9)\end{matrix}$

For further consideration, at first only a two-dimensionalrepresentation will be described for reasons of simplification. Here,some point on the original wavefront (r=o) or the propagated wavefront(r=p) is described by

$\begin{matrix}{{w_{r}(y)} = \begin{pmatrix}y \\{w_{r}(y)}\end{pmatrix}} & (10)\end{matrix}$where w_(r)(y) is described by:

$\begin{matrix}{{w_{r}(y)} = {\sum\limits_{k = 0}^{\infty}\;{\frac{a_{r,k}}{k!}y^{k}}}} & (11)\end{matrix}$

The coefficients a_(o,k) of the original wavefront 18 correspond to thederivatives of the wavefront with y=0:

$\begin{matrix}{a_{o,k} = {\left. {\frac{\partial^{k}}{\partial y^{k}}{w_{o}(y)}} \right|_{y = 0} = {w_{o}^{(k)}(0)}}} & (12)\end{matrix}$

In two dimensions, the vergence matrix S_(o) in equation (5) is reducedto a scalar E_(o,k)=nw_(o) ^((k))=na_(o,k). For second or third-orderaberrations, e.g. S_(o)=E_(o,2)=nw_(o) ⁽²⁾=na_(o,2), E_(o,3)=nw_(o)^((k))=na_(o,3), etc. result. The same applies to the propagatedwavefront 20.

Here, it is to be noted that any wavefront at the intersection pointwith the main ray 10 is not inclined with respect to the z axis. Sincethe z axis points along the direction of the main ray 10, it isperpendicular to the original and propagated wavefronts in theintersection points of the main ray 10 with the wavefronts 18, 20.Moreover, since the origin of the coordinate system is at the originalwavefront 18, it holds for the coefficients that: a_(o,0)=0, a_(o,1)=0,a_(p,0)=d, and a_(p,1)=0

In two dimensions, the normal vector n_(w)(y) for a wavefront w(y)results from n_(w)(y)=(−w⁽¹⁾(y),1)^(T)/√{square root over (1+w⁽¹⁾(y)²)},Where w⁽¹⁾=∂w/∂y. For Reasons of a simplified notation, first of allv≡w⁽¹⁾ and the following function is introduced:

$\begin{matrix}{{n(v)}:={\frac{1}{\sqrt{1 + v^{2}}}\begin{pmatrix}{- v} \\1\end{pmatrix}}} & (13)\end{matrix}$

As derivatives n^((i))(0)≡∂^(i)/∂v^(i) n(v)|_(v=0) of this functionthere result:

$\begin{matrix}{{{n(0)}:=\begin{pmatrix}0 \\1\end{pmatrix}},{{n^{(1)}(0)}:=\begin{pmatrix}{- 1} \\0\end{pmatrix}},{{n^{(2)}(0)}:=\begin{pmatrix}0 \\{- 1}\end{pmatrix}},{{n^{(3)}(0)}:=\begin{pmatrix}3 \\0\end{pmatrix}},{{n^{(4)}(0)}:=\begin{pmatrix}0 \\9\end{pmatrix}},{{etc}.}} & (14)\end{matrix}$

The normal vector, which is perpendicular to both the original wavefront18 and the propagated wavefront 20, can be designated uniformly withn_(w). Thus, for the first derivative of the normal vector there isdetermined:

$\begin{matrix}{\left. {\frac{\partial}{\partial y}{n_{w}(y)}} \middle| {}_{y = 0}{\equiv {n_{w}^{(1)}(0)}} \right. = {{{n^{(1)}(0)}{w_{o}^{(2)}(0)}} = {\begin{pmatrix}{- 1} \\0\end{pmatrix}{w_{o}^{(2)}(0)}}}} & (15)\end{matrix}$

The same applies to the higher derivatives.

With the local aberrations of the original wavefront 18, thecorresponding coefficients a_(k) and, equivalent thereto, thederivatives of the wavefront are directly defined as well. Subsequently,the propagated wavefront 20 is determined therefrom particularly bydetermining its derivatives or coefficients a_(k) for all orders 2≦k≦k₀up to the desired value k₀, and thus the values of the local aberrationsof the propagated wavefront 20 are determined.

As a starting point, the following situation with respect to FIG. 4 willbe considered in an illustrative way. While the main ray 10 and thecoordinate system are fixed, a neighboring ray 24 scans the originalwavefront 18 ({w_(o)}) and strikes it in a section y_(o)≠0. From there,it propagates further to the propagated wavefront 20 ({w_(p)}). Asillustrated in FIG. 4, y_(o) designates the projection of theintersection point of the neighboring ray 24 with the original wavefront{w_(o)} to the y axis, while analogously the projection of theintersection point with the propagated wavefront {w_(p)} to the y axisis designated with y_(p).

The vector w_(o)=w_(o)(y_(o)) (cf. equation (10)) points to theintersection point of the neighboring ray 24 with the original wavefront18, and the optical path difference (OPD) with respect to the propagatedwavefront 20 is designated with τ. Accordingly, the vector from theoriginal wavefront 18 to the propagated surface 20 is represented by τ/nn_(w). Thus, it results for the vector to the corresponding point of thepropagated wavefront: w_(p)=w_(o)+τ/n n_(w). As a basic equation thereis introduced:

$\begin{matrix}{{\begin{pmatrix}y_{o} \\{w_{o}\left( y_{o} \right)}\end{pmatrix} + {\frac{\tau}{n}n_{w}}} = \begin{pmatrix}y_{p} \\{w_{p}\left( y_{p} \right)}\end{pmatrix}} & (16)\end{matrix}$

Now, from this equation, the desired relations are derived order byorder. Here, y_(p) is preferably used as a free variable, on which y_(o)depends in turn. For solving the equation, first of all the vector

$\begin{matrix}{{p\left( y_{p} \right)} = \begin{pmatrix}{y_{o}\left( y_{p} \right)} \\{w_{p}\left( y_{p} \right)}\end{pmatrix}} & (17)\end{matrix}$on the boundary condition

${p(0)} = \begin{pmatrix}0 \\\frac{\tau}{n}\end{pmatrix}$can be introduced. Based on this, the following function is introducedfor the further consideration:

$\begin{matrix}{{f\left( {p,y_{p}} \right)} = \begin{pmatrix}{y_{o} + {\frac{\tau}{n}{n_{w,y}\left( {w_{o}^{(1)}\left( y_{o} \right)} \right)}} - y_{p}} \\{{w_{o}\left( y_{o} \right)} + {\frac{\tau}{n}{n_{w,z}\left( {w_{o}^{(1)}\left( y_{o} \right)} \right)}} - w_{p}}\end{pmatrix}} & (18)\end{matrix}$where (p₁,p₂)=(y_(o),w_(p)) are the components of p. Now, if p=p(y_(p)),the equation (16) can be represented in a compact form by:f(p(y _(p))y _(p))=0  (19)

The derivatives of this function according to y_(p) are preferablyexpressed by the following system of differential equations:

$\begin{matrix}{{{{\sum\limits_{j = 1}^{2}\;{\frac{\partial f_{i}}{\partial p_{j}}{p_{j}^{(1)}\left( y_{p} \right)}}} + \frac{\partial f_{i}}{\partial y_{p}}} = 0},{i = 1.2}} & (20)\end{matrix}$where the matrix with the elements A_(ij):=∂ƒ_(i)/∂p_(j) is referred toas a Jacobi matrix A. The Jacobi matrix A thus reads

$\begin{matrix}{{A\text{:} = \begin{pmatrix}\frac{\partial f_{1}}{\partial y_{o}} & \frac{\partial f_{1}}{\partial w_{p}} \\\frac{\partial f_{2}}{\partial y_{o}} & \frac{\partial f_{2}}{\partial w_{p}}\end{pmatrix}} = \begin{pmatrix}{1 + {\frac{\tau}{n}n_{w,y}^{(1)}w_{o}^{(2)}}} & 0 \\{w_{o}^{(1)} + {\frac{\tau}{n}n_{w,z}^{(1)}w_{o}^{(2)}}} & {- 1}\end{pmatrix}} & (21)\end{matrix}$

The terms appearing in this equation are to be understood as w_(o)⁽¹⁾≡w_(o) ⁽¹⁾(y_(o)) w_(o) ⁽²⁾≡w_(o) ⁽²⁾(y_(o)), n_(w,y)≡n_(w,y)(w_(o)⁽¹⁾(y_(o))), n_(w,y) ⁽¹⁾≡n_(w,y) ⁽¹⁾(w_(o) ⁽¹⁾(y_(o))), etc., wherey_(o),w_(p) are in turn themselves functions of y_(p).

The derivative vector ∂ƒ_(i)/∂y_(p) can be summarized as

$\begin{matrix}{{b\text{:} = \frac{\partial f}{\partial y_{p}}} = \begin{pmatrix}1 \\0\end{pmatrix}} & (22)\end{matrix}$

Thus, the above differential equation system can be represented as:A(p(y _(p)))p ⁽¹⁾(y _(p))=b  (23)

Formally, this equation is solved by:p ⁽¹⁾(y _(p))=A(p(y _(p)))⁻¹ b  (24)with the boundary condition

${p(0)} = {\begin{pmatrix}0 \\\frac{\tau}{n}\end{pmatrix}.}$Based on this, the equation system for higher-order aberrations ispreferably solved recursively as follows:

$\begin{matrix}{{{p^{(1)}(0)} = {A^{- 1}b}}{{p^{(2)}(0)} = {\left( A^{- 1} \right)^{(1)}b}}\ldots{{{p^{(k)}(0)} = {\left( A^{- 1} \right)^{({k - 1})}b}},}} & (25)\end{matrix}$with the abbreviatory designations A⁻¹=A(p(0))⁻¹=A(0)⁻¹

$\left( A^{- 1} \right)^{(1)} = {\frac{\mathbb{d}\;}{\mathbb{d}y_{p}}{A\left( {p\left( y_{p} \right)} \right)}^{- 1}{{_{y_{p} = 0}{,\ldots\mspace{11mu},{\left( A^{- 1} \right)^{({k - 1})} = {\frac{\mathbb{d}^{k - 1}}{\mathbb{d}y_{p}^{k - 1}}{A\left( {p\left( y_{p} \right)} \right)}^{- 1}}}}}_{y_{p} = 0}.}}$

In an alternative approach, it is suggested performing the recursion onthe basis of equation (23) instead of equation (24). The first (k−1)derivatives of equation (23) yield:

$\begin{matrix}{\mspace{79mu}{\begin{matrix}{{{Ap}^{(1)}(0)} = b} & (a) \\{{{A^{(1)}{p^{(1)}(0)}} + {{Ap}^{(2)}(0)}} = 0} & (b) \\{{{A^{(2)}{p^{(1)}(0)}} + {2A^{(1)}{p^{(2)}(0)}} + {{Ap}^{(3)}(0)}} = 0} & (c) \\\ldots & \; \\{{{\sum\limits_{j = 1}^{k}\;{\begin{pmatrix}{k - 1} \\{j - 1}\end{pmatrix}\; A^{({k - j})}{p^{(j)}(0)}}} = 0},{k \geq 2}} & (d)\end{matrix}\mspace{79mu}{where}\mspace{79mu}{{A = {{A\left( {p(0)} \right)} = {A(0)}}},{A^{(1)} = {\frac{\mathbb{d}}{\mathbb{d}y_{p}}{A\left( {p\left( y_{p} \right)} \right)}{_{y_{p} = 0}{,\ldots\mspace{11mu},{A^{({k - j})} = {\frac{\mathbb{d}^{k - j}}{\mathbb{d}y_{p}^{k - j}}{A\left( {p\left( y_{p} \right)} \right)}}}}}_{y_{p} = 0}}}}}} & (26)\end{matrix}$designate the total derivatives of the function. Formally, theseequations are solved by:

$\begin{matrix}\begin{matrix}{{{p^{(1)}(0)} = {A^{- 1}b}},} & {k = 1} \\{{{p^{(k)}(0)} = {{- A^{- 1}}{\sum\limits_{j = 1}^{k - 1}\;{\begin{pmatrix}{k - 1} \\{j - 1}\end{pmatrix}A^{({k - j})}{p^{(j)}(0)}}}}},} & {k \geq 2.}\end{matrix} & (27)\end{matrix}$

In order to obtain A(0)⁻¹, preferably equation (21) is evaluated for p=0and equation (14) is applied. This yields:

$\begin{matrix}{{A(0)} = {\left. \begin{pmatrix}{1 + {\frac{\tau}{n}w_{o}^{(2)}}} & 0 \\0 & {- 1}\end{pmatrix}\Rightarrow{A(0)}^{- 1} \right. = \begin{pmatrix}\frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}}} & 0 \\0 & {- 1}\end{pmatrix}}} & (28)\end{matrix}$from which it results for p⁽¹⁾(0):

$\begin{matrix}{{p^{(1)}(0)} = {{A^{- 1}b} = \begin{pmatrix}\frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}}} \\0\end{pmatrix}}} & (29)\end{matrix}$

In turn, this means

${y_{o}^{(1)}(0)} = {{\frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}}}\mspace{14mu}{and}\mspace{14mu}{w_{p}^{(1)}(0)}} = 0.}$For orders k≧2, preferably equation (27) is applied. The derivatives

${A^{(1)} = {{\frac{\mathbb{d}}{\mathbb{d}y_{p}}{A\left( {p\left( y_{p} \right)} \right)}}❘_{y_{p} = 0}}},$etc. are preferably determined from equation (21) and preferablyequation (14) is applied again. Thus, it results in the second order:

$\begin{matrix}{w_{p}^{(2)} = {\frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}}}w_{o}^{(2)}}} & (30)\end{matrix}$which basically corresponds to the above-described propagation equation.The higher orders can analogously be expressed by:

$\begin{matrix}{\mspace{79mu}{{{{w_{p}^{(3)} = {\left( \frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}}} \right)^{3}w_{o}^{(3)}}}\mspace{20mu}{w_{p}^{(4)} = {\left( \frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}}} \right)^{4}\left( {w_{o}^{(4)} + {3\frac{\tau}{n}\left( {{\frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}}}w_{o}^{{(3)}^{2}}} - w_{o}^{{(2)}^{4}}} \right)}} \right)}}w_{p}^{(5)}} = {\left( \frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}}} \right)^{5}\left( {w_{o}^{(5)} + {5\frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}}}\frac{\tau}{n}{w_{o}^{(3)}\left( {{2\; w_{o}^{(4)}} + {3\frac{1}{1 - {\frac{\tau}{n}w_{o}^{(2)}n}}\frac{\tau}{n}w_{o}^{{(3)}^{2}}} - {6\; w_{o}^{{(2)}^{3}}}} \right)}}} \right)}}\mspace{79mu}\ldots}} & (31)\end{matrix}$

Equation (31) correspondingly applies to the derivatives and thecoefficients a_(o,k) and a_(p,k) due to equations (10) to (12). Now, ifone replaces d=τ/n and

${\beta = \frac{1}{1 - {\frac{d}{n}S_{o}}}},$the local aberrations can be expressed as follows:

$\begin{matrix}{\mspace{79mu}{{{S_{p} = {\beta\; S_{o}}}\mspace{20mu}{E_{p,3} = {\beta^{3}E_{o,3}}}\mspace{20mu}{E_{p,4} = {\beta^{4}\left( {E_{o,4} + {3\frac{d}{n}\left( {{\beta\; E_{o,3}^{2}} - \frac{S_{o}^{4}}{n^{2}}} \right)}} \right)}}\mspace{20mu}{E_{p,5} = {\beta^{5}\left( {E_{o,5} + {5\beta\frac{d}{n}{E_{o,3}\left( {{2\; E_{o,4}} + {3\beta\frac{d}{n}E_{o,3}^{2}} - {6\frac{S_{o}^{3}}{n^{2}}}} \right)}}} \right)}}E_{p,6}} = {\beta^{6}\left( {E_{o,6} + {5\beta\frac{d}{n}\left( {{3\; E_{o,3}E_{o,5}} + {21\beta\frac{d}{n}E_{o,3}^{2}E_{o,4}} - {12\frac{S_{o}^{3}E_{o,4}}{n^{2}}} + {2\; E_{o,4}} - {9\beta\; S_{0}^{2}E_{o,3}^{2}\frac{3 + {4\frac{d}{n}S_{o}}}{n^{2}}} + {21\left( {\beta\frac{d}{n}} \right)^{2}E_{o,3}^{4}} + {9\; S_{o}^{6}\frac{3 + {\frac{d}{n}S_{o}}}{n^{4}}}} \right)}} \right)}}} & (32)\end{matrix}$

For 2<k≦6, this is preferably represented byE _(p,k)=β^(k)(E _(o,k) +R _(k))  (33)in a generalized way, where in R_(k) all wavefront derivatives E_(o,j)of the lower orders (j<k) are expressed in form of local aberrations.

Even if a three-dimensional representation is more complex, it canbasically be established by analogy with the two-dimensionalrepresentation. Therefore, for the fully three-dimensionalrepresentation, only a few essential additional considerations will bedescribed in the following.

Preferably, the original wavefront can be expressed by the 3D vector

$\begin{matrix}{{w_{o}\left( {x,y} \right)} = \begin{pmatrix}x \\y \\{w_{o}\left( {x,y} \right)}\end{pmatrix}} & (34)\end{matrix}$

where w_(o) (x, y) is determined according to equation (2), and therelationship between the coefficients and the derivatives is determinedaccording to equation (3). The connection between the coefficients andthe local aberrations results from a multiplication of the coefficientby the refractive index. Preferably, by analogy with equation (13),formal vectors are introduced:

$\begin{matrix}{{n\left( {u,v} \right)}:={\frac{1}{\sqrt{1 + u^{2} + v^{2}}}\begin{pmatrix}{- u} \\{- v} \\1\end{pmatrix}}} & (35)\end{matrix}$so that the normal vectors with respect to a surfacew(x,y):=(x,y,w(x,y))^(T) are determined by:

$\frac{w^{({1,0})} \times w^{({0,1})}}{{w^{({1,0})} \times w^{({0,1})}}} = {{\frac{1}{\sqrt{1 + w^{{({1,0})}^{2} + w^{{({0,1})}^{2}}}}}\begin{pmatrix}{- w^{({1,0})}} \\{- w^{({0,1})}} \\1\end{pmatrix}} = {{n\left( {w^{({1,0})},w^{({0,1})}} \right)} = {n\left( {\nabla w} \right)}}}$

In the intersection point, it thus results n_(w)(0,0)=(0,0,1)^(T), andthe derivatives according to equation (14) are preferably determinedfrom equation (35).

As the basis for the consideration of a connection between the originaland propagated wavefronts, preferably substantially equation (16) isused, with the difference that now x and y components are considered atthe same time. As a vector of unknown functions, there is preferablydetermined:

$\begin{matrix}{{p\left( {x_{p},y_{p}} \right)} = \begin{pmatrix}{x_{o}\left( {x_{p},y_{p}} \right)} \\{y_{o}\left( {x_{p},y_{p}} \right)} \\{w_{p}\left( {x_{p},y_{p}} \right)}\end{pmatrix}} & (36)\end{matrix}$and by analogy with equation (16), there is preferably used for thethree-dimensional consideration:f(p(x _(p) ,y _(p))x _(p) ,y _(p))=0  (37)where f is analogous to equation (18).

An importance difference compared to the two-dimensional considerationis that in the three-dimensional case two arguments exist, with respectto which the derivatives are taken into account. Thus, already in thefirst order, two equations are considered:A(p(x _(p) ,y _(p)))p ^((1,0))(x _(p) ,y _(p))=b _(x)A(p(x _(p) ,y _(p)))p ^((0,1))(x _(p) ,y _(p))=b _(y)  (38)where the inhomogeneity is described by the column vectors:

$\begin{matrix}{{b_{x} = {{- \frac{\partial f}{\partial x_{p}}} = \begin{pmatrix}1 & 0 & 0\end{pmatrix}^{T}}},{b_{y} = {{- \frac{\partial f}{\partial y_{p}}} = \begin{pmatrix}0 & 1 & 0\end{pmatrix}^{T}}}} & (39)\end{matrix}$

The Jacobi matrix A(p(x₂,y₂)) with the elements A_(ij):=∂ƒ_(i)/∂p_(j) isthe same for both equations and analogous to equation (21), but now inthe size 3×3.

$\begin{matrix}{{A\left( {p\left( {x_{p},y_{p}} \right)} \right)} = \begin{pmatrix}\begin{matrix}{1 + {\frac{\tau}{n}\left( {{n_{w,x}^{({0,1})}w_{o}^{({1,1})}} +} \right.}} \\\left. {n_{w,x}^{({1,0})}w_{o}^{({2,0})}} \right)\end{matrix} & {\frac{\tau}{n}\left( {{n_{w,x}^{({0,1})}w_{o}^{({0,2})}} + {n_{w,x}^{({1,0})}w_{o}^{({1,1})}}} \right)} & 0 \\{\frac{\tau}{n}\left( {{n_{w,y}^{({0,1})}w_{o}^{({1,1})}} + {n_{w,y}^{({1,0})}w_{o}^{({2,0})}}} \right)} & {1 + {\frac{\tau}{n}\left( {{n_{w,y}^{({0,1})}w_{o}^{({0,2})}} + {n_{w,y}^{({1,0})}w_{o}^{({1,1})}}} \right)}} & 0 \\\begin{matrix}{w_{o}^{({1,0})} +} \\{\frac{\tau}{n}\left( {{n_{w,z}^{({0,1})}w_{o}^{({1,1})}} + {n_{w,z}^{({1,0})}w_{o}^{({2,0})}}} \right)}\end{matrix} & \begin{matrix}{w_{o}^{({0,1})} +} \\{\frac{\tau}{n}\left( {{n_{w,z}^{({0,1})}w_{o}^{({0,2})}} + {n_{w,z}^{({1,0})}w_{o}^{({1,1})}}} \right)}\end{matrix} & {- 1}\end{pmatrix}} & (40)\end{matrix}$

The direct solutions by analogy with equation (25) are now determined by

$\begin{matrix}{\mspace{79mu}{{{p^{({1,0})}\left( {0,0} \right)} = {A^{- 1}b_{x}}}\mspace{20mu}{{p^{({0,1})}\left( {0,0} \right)} = {A^{- 1}b_{y}}}\mspace{20mu}{{p^{({2,0})}\left( {0,0} \right)} = \left. \left( A^{- 1} \right)^{({1,0})} \middle| b_{x} \right.}\mspace{20mu}{{p^{({1,1})}\left( {0,0} \right)} = {{\left( A^{- 1} \right)^{({0,1})}b_{x}} = \left. \left( A^{- 1} \right)^{({1,0})} \middle| b_{y} \right.}}\mspace{20mu}{{p^{({0,2})}\left( {0,0} \right)} = {\left( A^{- 1} \right)^{({0,1})}b_{y}}}\mspace{20mu}\ldots{{p^{({k_{x},k_{y}})}\left( {0,0} \right)} = \left\{ {{{\begin{matrix}{{\left( A^{- 1} \right)^{({{k_{x} - 1},0})}b_{x}},} & {{k_{x} \neq 0},{k_{y} = 0}} \\{{{\left( A^{- 1} \right)^{({{k_{x} - 1},k_{y}})}b_{x}} = {\left( A^{- 1} \right)^{({k_{x},{k_{y} - 1}})}b_{y}}},} & {{k_{x} \neq 0},{k_{y} \neq 0}} \\{{\left( A^{- 1} \right)^{({0,{k_{y} - 1}})}b_{y}},} & {{k_{x} = 0},{k_{y} \neq 0}}\end{matrix}\mspace{20mu}{where}\mspace{14mu} A^{- 1}} = {{A\left( {p\left( {0,0} \right)} \right)}^{- 1} = {A(0)}^{- 1}}},\mspace{20mu}{\left( A^{- 1} \right)^{({1,0})} = \left. {\frac{\mathbb{d}}{\mathbb{d}x_{p}}{A\left( {p\left( {x_{p},y_{p}} \right)} \right)}^{- 1}} \right|_{{x_{p} = 0},{y_{p} = 0}}},\mspace{20mu}{\left( A^{- 1} \right)^{({k_{x},k_{y}})} = \left. {\frac{\mathbb{d}^{k_{x}}}{\mathbb{d}x_{p}^{k_{x}}}\frac{\mathbb{d}^{k_{y}}}{\mathbb{d}x_{p}^{k_{y}}}{A\left( {p\left( {x_{p},y_{p}} \right)} \right)}^{- 1}} \right|_{{x_{p} = 0},{y_{p} = 0}}},{{etc}..}} \right.}}} & (41)\end{matrix}$

By analogy with equations (28) and (29), it results for thethree-dimensional consideration:

$\begin{matrix}{{{A(0)} = {\left. \begin{pmatrix}{1 - {\frac{\tau}{n}w_{o}^{({2,0})}}} & {{- \frac{\tau}{n}}w_{o}^{({1,1})}} & 0 \\{{- \frac{\tau}{n}}w_{o}^{({1,1})}} & {1 - {\frac{\tau}{n}w_{o}^{({0,2})}}} & 0 \\0 & 0 & {- 1}\end{pmatrix}\Rightarrow{A(0)}^{- 1} \right. = \begin{pmatrix}{\gamma\begin{pmatrix}{1 - {\frac{\tau}{n}w_{o}^{({0,2})}}} & {\frac{\tau}{n}w_{o}^{({1,1})}} \\{\frac{\tau}{n}w_{o}^{({1,1})}} & {1 - {\frac{\tau}{n}w_{o}^{({2,0})}}}\end{pmatrix}} & \; & \begin{matrix}0 \\0\end{matrix} \\\begin{matrix}0 & \; & \; & \; & \; & 0\end{matrix} & \; & {- 1}\end{pmatrix}}}\mspace{20mu}{with}{\gamma = {\frac{- 1}{\det\left( {A(0)} \right)} = \frac{1}{\left. {1 - {\frac{\tau}{n}w_{o}^{({2,0})}} - \left( {\frac{\tau}{n}w_{o}^{({1,1})}} \right)^{2} - {\frac{\tau}{n}w_{o}^{({0,2})}} + {\left( \frac{\tau}{n} \right)^{2}w_{o}^{({2,0})}w_{o}^{({0,2})}}} \right)}}}} & (42)\end{matrix}$and after application of equations (39) and (41), the solutions

$\begin{matrix}{{{p^{({1,0})}\left( {0,0} \right)} = {\gamma\begin{pmatrix}{n\left( {n - {\tau\; w_{o}^{({0,2})}}} \right)} \\{n\;\tau\; w_{o}^{({1,1})}} \\0\end{pmatrix}}},{{p^{({0,1})}\left( {0,0} \right)} = {\gamma\begin{pmatrix}{n\;\tau\; w_{o}^{({1,1})}} \\{n\left( {n - {\tau\; w_{o}^{({2,0})}}} \right)} \\0\end{pmatrix}}}} & (43)\end{matrix}$

After further application of equations (39) and (41), it results in thesecond orderw _(p) ^((2,0))=γ(τ/n(w _(o) ^((1,1)))²+(1−τ/n w _(o) ^((0,2)))w _(o)^((2,0)))w _(p) ^((1,1)) =γw _(o) ^((1,1))w _(p) ^((0,2))=γ(τ/n(w _(o) ^((1,1)))²+(1−τ/n w _(o) ^((2,0)))w _(o)^((0,2)))  (44)

In a preferred embodiment, the coordinate axes for determination of thepropagation are selected or determined such that the x axis and the yaxis coincide with the directions of the main curvatures of the originalwavefront. It thereby holds that w_(o) ^((1,1))=0, and the equations(44) are simplified as

$\begin{matrix}{{w_{p}^{({2,0})} = {\frac{1}{1 - {\frac{\tau}{n}w_{o}^{({2,0})}}}w_{o}^{({2,0})}}}{w_{p}^{({1,1})} = 0}{w_{p}^{({0,2})} = {\frac{1}{1 - {\frac{\tau}{n}w_{o}^{({0,2})}}}w_{o}^{({0,2})}}}} & (45)\end{matrix}$

In a corresponding way, the equations in the third order are preferablyexpressed as follows:

$\begin{matrix}{{w_{p}^{({3,0})} = {\gamma^{3}\left( {{\left( {1 - {\frac{\tau}{n}w_{o}^{({0,2})}}} \right)^{3}w_{o}^{({3,0})}} + {\frac{\tau}{n}{w_{o}^{({1,1})}\left( {{3\left( {1 - {\frac{\tau}{n}w_{o}^{({0,2})}}} \right)^{2}w_{o}^{({2,1})}} + {\frac{\tau}{n}{w_{o}^{({1,1})}\left( {\frac{\tau}{n}w_{o}^{({0,3})}w_{o}^{({1,1})}} \right)}} + {3\left( {1 - {\frac{\tau}{n}w_{o}^{({0,2})}}} \right)^{2}w_{o}^{({2,1})}}} \right)}}} \right)}}{w_{p}^{({2,1})} = {\gamma^{3}\left( {w_{o}^{({2,1})} + {\frac{\tau}{n}\left( {{w_{o}^{({1,1})}\left( {{2w_{o}^{({1,2})}} + w_{o}^{({3,0})}} \right)} - {\left( {{2w_{o}^{({0,2})}} + w_{o}^{({2,0})}} \right)w_{o}^{({2,1})}}} \right)} + {\left( \frac{\tau}{n} \right)^{2}\left( {{w_{o}^{({2,1})}w_{o}^{{({0,2})}^{2}}} - {2\left( {{w_{o}^{({1,1})}\left( {w_{o}^{({1,2})} + w_{o}^{({3,0})}} \right)} - {w_{o}^{({2,0})}w_{o}^{({2,1})}}} \right)w_{o}^{({0,2})}} + {w_{o}^{({0,3})}w_{o}^{{({1,1})}^{2}}} + {2\;{w_{o}^{({1,1})}\left( {{w_{o}^{({1,1})}w_{o}^{({2,1})}} - {w_{o}^{({1,2})}w_{o}^{({2,0})}}} \right)}}} \right)} + {\left( \frac{\tau}{n} \right)^{3}\left( {{w_{o}^{({1,2})}w_{o}^{{({1,1})}^{3}}} - {\left( {{w_{o}^{({0,3})}w_{o}^{({2,0})}} + {2\; w_{o}^{({0,2})}w_{o}^{({2,1})}}} \right)w_{o}^{{({1,1})}^{2}}} + {{w_{o}^{({0,2})}\left( {{2\; w_{o}^{({1,2})}w_{o}^{({2,0})}} + {w_{o}^{({0,2})}w_{o}^{({3,0})}}} \right)}w_{o}^{({1,1})}} - {w_{o}^{{({0,2})}^{2}}w_{o}^{({2,0})}w_{o}^{({2,1})}}} \right)}} \right)}}{w_{p}^{({1,2})} = {\gamma^{3}\left( {w_{o}^{({1,2})} + {\frac{\tau}{n}\left( {{w_{o}^{({1,1})}\left( {{2w_{o}^{({2,1})}} + w_{o}^{({0,3})}} \right)} - {\left( {{2w_{o}^{({2,0})}} + w_{o}^{({0,2})}} \right)w_{o}^{({1,2})}}} \right)} + {\left( \frac{\tau}{n} \right)^{2}\left( {{w_{o}^{({1,2})}w_{o}^{{({2,0})}^{2}}} - {2\left( {{w_{o}^{({1,1})}\left( {w_{o}^{({2,1})} + w_{o}^{({0,3})}} \right)} - {w_{o}^{({0,2})}w_{o}^{({1,2})}}} \right)w_{o}^{({2,0})}} + {w_{o}^{({3,0})}w_{o}^{{({1,1})}^{2}}} + {2\;{w_{o}^{({1,1})}\left( {{w_{o}^{({1,1})}w_{o}^{({1,2})}} - {w_{o}^{({2,1})}w_{o}^{({0,2})}}} \right)}}} \right)} + {\left( \frac{\tau}{n} \right)^{3}\left( {{w_{o}^{({2,1})}w_{o}^{{({1,1})}^{3}}} - {\left( {{w_{o}^{({3,0})}w_{o}^{({0,2})}} + {2\; w_{o}^{({2,0})}w_{o}^{({1,2})}}} \right)w_{o}^{{({1,1})}^{2}}} + {{w_{o}^{({2,0})}\left( {{2\; w_{o}^{({2,1})}w_{o}^{({0,2})}} + {w_{o}^{({2,0})}w_{o}^{({0,3})}}} \right)}w_{o}^{({1,1})}} - {w_{o}^{{({2,0})}^{2}}w_{o}^{({0,2})}w_{o}^{({1,2})}}} \right)}} \right)}}{w_{p}^{({0,3})} = {\gamma^{3}\left( {{\left( {1 - {\frac{\tau}{n}w_{o}^{({2,0})}}} \right)^{3}w_{o}^{({0,3})}} + {\frac{\tau}{n}{w_{o}^{({1,1})}\left( {{3\left( {1 - {\frac{\tau}{n}w_{o}^{({2,0})}}} \right)^{2}w_{o}^{({1,2})}} + {\frac{\tau}{n}{w_{o}^{({1,1})}\left( {\frac{\tau}{n}w_{o}^{({3,0})}w_{o}^{({1,1})}} \right)}} + {3\left( {1 - {\frac{\tau}{n}w_{o}^{({2,0})}}} \right)^{2}w_{o}^{({2,1})}}} \right)}}} \right)}}} & (46)\end{matrix}$

Now, if one replaces

${d = {{\frac{\tau}{n}\mspace{14mu}{and}\mspace{14mu}\gamma} = \frac{1}{\left. {1 - {\frac{d}{n}S_{oxx}} - \left( {\frac{d}{n}S_{oxy}} \right)^{2} - {\frac{d}{n}S_{oyy}} + {\left( \frac{d}{n} \right)^{2}S_{oxx}S_{oyy}}} \right)}}},$the propagation of the wavefront in the second order in the form of thelocal aberrations can be expressed as follows:

$\begin{matrix}{s_{p} = {\gamma\left( {s_{o} + {\frac{d}{n}\begin{pmatrix}{S_{oxy}^{2} - {S_{oxx}S_{oyy}}} \\0 \\{S_{oxy}^{2} - {S_{oxx}S_{oyy}}}\end{pmatrix}}} \right)}} & (47)\end{matrix}$

Moreover, if one replaces

${\beta_{x} = {{\frac{1}{1 - {\frac{d}{n}S_{xx}}}\mspace{14mu}{and}\mspace{14mu}\beta_{y}} = \frac{1}{1 - {\frac{d}{n}S_{yy}}}}},$the propagation of the wavefront in the third order can be described by:

$\begin{matrix}{e_{p\; 3} = {{\gamma^{3}\begin{pmatrix}\beta_{y}^{- 3} & {3\beta_{y}^{- 2}\frac{d}{n}S_{xy}} & {3{\beta_{y}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2}} & \left( {\frac{d}{n}S_{xy}} \right)^{3} \\{\beta_{y}^{- 2}\frac{d}{n}S_{xy}} & {\beta_{y}^{- 1}\left( {\frac{1}{\gamma} + {3\left( {\frac{d}{n}S_{xy}} \right)^{2}}} \right)} & {{\frac{2}{\gamma}\frac{d}{n}S_{xy}} + {3\left( {\frac{d}{n}S_{xy}} \right)^{3}}} & {\beta_{x}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2} \\{\beta_{y}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2} & {{\frac{2}{\gamma}\frac{d}{n}S_{xy}} + {3\left( {\frac{d}{n}S_{xy}} \right)^{3}}} & {\beta_{x}^{- 1}\left( {\frac{1}{\gamma} + {3\left( {\frac{d}{n}S_{xy}} \right)^{2}}} \right)} & {\beta_{x}^{- 2}\frac{d}{n}S_{xy}} \\\left( {\frac{d}{n}S_{xy}} \right)^{3} & {2{\beta_{x}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2}} & {3\beta_{x}^{- 2}\frac{d}{n}S_{xy}} & \beta_{x}^{- 3}\end{pmatrix}}e_{o\; 3}}} & (48)\end{matrix}$

In a preferred embodiment, the coordinate axes for determination of thepropagation are selected or determined such that the x axis and the yaxis coincide with the directions of the main curvatures of the originalwavefront. Thereby, the equations (47) and (48) are simplified as

$\begin{matrix}{s_{p} = {\begin{pmatrix}\beta_{x} & 0 & 0 \\0 & 0 & 0 \\0 & 0 & \beta_{y}\end{pmatrix}s_{o}}} & (49) \\{e_{p\; 3} = {\begin{pmatrix}\beta_{x}^{3} & 0 & 0 & 0 \\0 & {\beta_{x}^{2}\beta_{y}} & 0 & 0 \\0 & 0 & {\beta_{x}\beta_{y}^{2}} & 0 \\0 & 0 & 0 & \beta_{y}^{3}\end{pmatrix}e_{o\; 3}}} & (50)\end{matrix}$

The propagation of fourth-order aberrations can be determined in acomparatively simply way by:

$\begin{matrix}{e_{p\; 4} = {\begin{pmatrix}\beta_{x}^{4} & \ldots & \; & \ldots & 0 \\\vdots & {\beta_{x}^{3}\beta_{y}^{1}} & \; & \; & \vdots \\\; & \; & {\beta_{x}^{2}\beta_{y}^{2}} & \; & \; \\\vdots & \; & \; & {\beta_{x}^{1}\beta_{y}^{3}} & \vdots \\0 & \; & \; & \ldots & \beta_{y}^{4}\end{pmatrix}\left( {e_{o\; 4} + {\frac{d}{n}\begin{pmatrix}{3\left( {{\beta_{x}E_{oxxx}^{2}} + {\beta_{y}E_{oxxy}^{2}} - \frac{S_{oxx}^{4}}{n^{2}}} \right)} \\{3{E_{oxxy}\left( {{\beta_{x}E_{oxxx}} + {\beta_{y}E_{oxyy}}} \right)}} \\{{\beta_{x}\left( {{2E_{oxxy}^{2}} + {E_{oxxx}E_{oxyy}}} \right)} + {\beta_{y}\left( {{2E_{oxyy}^{2}} + {E_{oxxy}E_{oyyy}}} \right)} -} \\\left( \frac{S_{oxx}S_{oyy}}{n} \right)^{2} \\{3{E_{oxyy}\left( {{\beta_{x}E_{oxxy}} + {\beta_{y}E_{oyyy}}} \right)}} \\{3\left( {{\beta_{x}E_{oxyy}^{2}} + {\beta_{y}E_{oyyy}^{2}} - \frac{S_{oyy}^{4}}{n^{2}}} \right)}\end{pmatrix}}} \right)}} & (51)\end{matrix}$

For 2<k≦4, this is preferably generalized by

$\begin{matrix}{e_{pk} = {B_{k}\left( {e_{ok} + r_{k}} \right)}} & (52) \\{{{with}\mspace{14mu} B_{k}} = \begin{pmatrix}\beta_{x}^{k} & \ldots & \; & \ldots & 0 \\\vdots & {\beta_{x}^{k - 1}\beta_{y}^{1}} & \; & \; & \vdots \\\; & \; & \ddots & \; & \; \\\vdots & \; & \; & {\beta_{x}^{1}\beta_{y}^{k - 1}} & \vdots \\0 & \; & \; & \ldots & \beta_{y}^{k}\end{pmatrix}} & (53)\end{matrix}$where r_(k) represents a vector in which by analogy with R_(k) inequation (33) all remainder terms R_(k) _(x) _(,k) _(y) are included.

In a preferred embodiment it holds:s _(p) =T ⁽²⁾({circumflex over (R)})T ⁽²⁾({circumflex over (β)}){tildeover ({circumflex over (s)})}_(p) S({circumflex over (R)})e _(p3) =T ⁽³⁾({circumflex over (R)})T ⁽³⁾({circumflex over (β)}){tildeover ({circumflex over (e)})}_(p3) S({circumflex over (R)})e _(p4) =T ⁽⁴⁾({circumflex over (R)})T ⁽⁴⁾({circumflex over (β)}){tildeover ({circumflex over (e)})}_(p4) S({circumflex over (R)})etc.  (54)where s_(p), e_(p3), e_(p4), . . . apply in every coordinate system andwhere

$\mspace{79mu}{\hat{\beta} = \begin{pmatrix}{\hat{\beta}}_{xx} & 0 \\0 & {\hat{\beta}}_{yy}\end{pmatrix}}$      with$\mspace{79mu}{{\hat{\beta}}_{xx} = \left( {1 - {\frac{\tau}{n}{\hat{w}}^{({2,0})}}} \right)^{- 1}}$$\mspace{79mu}{{\hat{\beta}}_{yy} = \left( {1 - {\frac{\tau}{n}{\hat{w}}^{({0,2})}}} \right)^{- 1}}$$\begin{pmatrix}{\hat{w}}^{({2,0})} \\{\hat{w}}^{({1,1})} \\{\hat{w}}^{({0,2})}\end{pmatrix} = {{\frac{1}{2}\left( {w^{({2,0})} + w^{({0,2})}} \right)\begin{pmatrix}1 \\0 \\1\end{pmatrix}} + {\frac{1}{2}\left( {w^{({2,0})} - w^{({0,2})}} \right)\sqrt{1 + \left( \frac{2w^{({1,1})}}{w^{({2,0})} - w^{({0,2})}} \right)^{2}}\begin{pmatrix}1 \\0 \\{- 1}\end{pmatrix}}}$is an auxiliary matrix, which can be referred to back to the matrix

$\beta = {\begin{pmatrix}\beta_{xx} & \beta_{xy} \\\beta_{xy} & \beta_{yy}\end{pmatrix} = \left( {1 - {\frac{\tau}{n}\begin{pmatrix}w_{o}^{({2,0})} & w_{o}^{({1,1})} \\w_{o}^{({1,1})} & w_{o}^{({0,2})}\end{pmatrix}}} \right)^{- 1}}$ by β̂ = R̂ β R̂⁻¹ where$\hat{R} = \begin{pmatrix}{\cos\mspace{11mu}\varphi} & {{- \sin}\mspace{11mu}\varphi} \\{\sin\mspace{11mu}\varphi} & {\cos\mspace{11mu}\varphi}\end{pmatrix}$is a rotation matrix, which transforms from the special system in whichthe x axis and the y axis coincide with the directions of the maincurvatures of the original wavefront into the general system. Here,

$\varphi = {\frac{1}{2}{arc}\;\tan\;\frac{2\beta_{xy}}{\beta_{yy} - \beta_{xx}}}$

In equation (54), the matrix

${S\left( \hat{R} \right)}\text{:} = \begin{pmatrix}\hat{R} & 0 \\0 & 1\end{pmatrix}$is used, and, further, T⁽¹⁾, T⁽²⁾, T⁽³⁾, T⁽⁴⁾ in equation (54) arematrix-like functions which assign the matrices

$\mspace{79mu}{{T^{(1)}(X)} = \begin{pmatrix}a & c \\b & d\end{pmatrix}}$ $\mspace{79mu}{{T^{(2)}(X)} = \begin{pmatrix}a^{2} & {2{ac}} & c^{2} \\{ab} & {{ad} + {bc}} & {cd} \\b^{2} & {2{bd}} & d^{2}\end{pmatrix}}$ $\mspace{79mu}{{T^{(3)}(X)} = \begin{pmatrix}a^{3} & {3a^{2}c} & {3{ac}^{2}} & c^{3} \\{a^{2}b} & {a\left( {{ad} + {2{bc}}} \right)} & {c\left( {{2{ad}} + {bc}} \right)} & {c^{2}d} \\{ab}^{2} & {b\left( {{2{ad}} + {bc}} \right)} & {d\left( {{ad} + {2{bc}}} \right)} & {cd}^{2} \\b^{3} & {3b^{2}d} & {3{bd}^{2}} & d^{3}\end{pmatrix}}$ ${T^{(4)}(X)} = \begin{pmatrix}a^{4} & {4a^{3}c} & {6a^{2}c^{2}} & {4{ac}^{3}} & c^{4} \\{a^{3}b} & {a^{2}\left( {{3{bc}} + {ad}} \right)} & {3{{ac}\left( {{bc} + {ad}} \right)}} & {c^{2}\left( {{bc} + {3{ad}}} \right)} & {c^{3}d} \\{a^{2}b^{2}} & {2{{ab}\left( {{bc} + {ad}} \right)}} & {{b^{2}c^{2}} + {4{abcd}} + {a^{2}d^{2}}} & {2{{cd}\left( {{bc} + {ad}} \right)}} & {c^{2}d^{2}} \\{ab}^{3} & {b^{2}\left( {{bc} + {3{ad}}} \right)} & {3{{bd}\left( {{bc} + {ad}} \right)}} & {d^{2}\left( {{3{bc}} + {ad}} \right)} & {cd}^{3} \\b^{4} & {4b^{3}d} & {6b^{2}d^{2}} & {4{bd}^{3}} & d^{4}\end{pmatrix}$to a predetermined matrix

$x = \begin{pmatrix}a & b \\c & d\end{pmatrix}$

For even higher orders, the matrices T^((n)) can be defined with n>4.Finally, as expressions for solutions on which the solutions for thepropagated wavefronts can be formed by the transformation in equation(54), there are predetermined for the order n=2

${{\hat{\overset{\sim}{s}}}_{p} = {\begin{pmatrix}{\hat{\overset{\sim}{w}}}_{p}^{({2,0})} \\{\hat{\overset{\sim}{w}}}_{p}^{({1,1})} \\{\hat{\overset{\sim}{w}}}_{p}^{({0,2})}\end{pmatrix} = {\begin{pmatrix}{{\hat{\beta}}_{11}^{- 1}{\hat{w}}_{o}^{({2,0})}} \\\begin{matrix}0 \\{{\hat{\beta}}_{22}^{- 1}{\hat{w}}_{o}^{({0,2})}}\end{matrix}\end{pmatrix} = {\begin{pmatrix}{\hat{w}}^{({2,0})} \\0 \\{\hat{w}}_{o}^{({0,2})}\end{pmatrix} - {\frac{\tau}{n}\begin{pmatrix}{\hat{w}}_{o}^{{({2,0})}^{2}} \\0 \\{\hat{w}}_{o}^{{({0,2})}^{2}}\end{pmatrix}}}}}},$for the order n=3

${\hat{\overset{\sim}{e}}}_{p\; 3} = {\begin{pmatrix}{\hat{\overset{\sim}{w}}}_{p}^{({3,0})} \\{\hat{\overset{\sim}{w}}}_{p}^{({2,1})} \\{\hat{\overset{\sim}{w}}}_{p}^{({1,2})} \\{\hat{\overset{\sim}{w}}}_{p}^{({0,3})}\end{pmatrix} = \begin{pmatrix}{\hat{w}}_{o}^{({3,0})} \\{\hat{w}}_{o}^{({2,1})} \\{\hat{w}}_{o}^{({1,2})} \\{\hat{w}}_{o}^{({0,3})}\end{pmatrix}}$and for the order n=4

${\hat{\overset{\sim}{e}}}_{p\; 4} = {\begin{pmatrix}{\hat{\overset{\sim}{w}}}_{p}^{({4,0})} \\{\hat{\overset{\sim}{w}}}_{p}^{({3,1})} \\{\hat{\overset{\sim}{w}}}_{p}^{({2,2})} \\{\hat{\overset{\sim}{w}}}_{p}^{({1,3})} \\{\hat{\overset{\sim}{w}}}_{p}^{({0,4})}\end{pmatrix} = {\begin{pmatrix}{\hat{w}}_{o}^{({4,0})} \\{\hat{w}}_{o}^{({3,1})} \\{\hat{w}}_{o}^{({2,2})} \\{\hat{w}}_{o}^{({1,3})} \\{\hat{w}}_{o}^{({0,4})}\end{pmatrix} + {\frac{\tau}{n}\begin{pmatrix}{3\left( {{{\hat{\beta}}_{11}{\hat{w}}_{o}^{{({3,0})}^{2}}} + {{\hat{\beta}}_{22}{\hat{w}}_{o}^{{({2,1})}^{2}}} - {\hat{w}}_{o}^{{({2,0})}^{4}}} \right)} \\{3\;{{\hat{w}}_{o}^{({2,1})}\left( {{{\hat{\beta}}_{11}{\hat{w}}_{o}^{({3,0})}} + {{\hat{\beta}}_{22}{\hat{w}}_{o}^{({1,2})}}} \right)}} \\\begin{matrix}{{{\hat{w}}_{o}^{({1,2})}\left( {{{\hat{\beta}}_{11}{\hat{w}}_{o}^{({3,0})}} + {2{\hat{\beta}}_{22}{\hat{w}}_{o}^{({1,2})}}} \right)} + {{\hat{w}}_{o}^{({2,1})}\left( {{2{\hat{\beta}}_{11}{\hat{w}}_{o}^{({2,1})}} + {{\hat{\beta}}_{22}{\hat{w}}_{o}^{({0,3})}}} \right)} -} \\{{\hat{w}}_{o}^{{({2,0})}^{2}}{\hat{w}}_{o}^{{({0,2})}^{2}}}\end{matrix} \\{3\;{{\hat{w}}_{o}^{({1,2})}\left( {{{\hat{\beta}}_{11}{\hat{w}}_{o}^{({2,1})}} + {{\hat{\beta}}_{22}{\hat{w}}_{o}^{({0,3})}}} \right)}} \\{3\left( {{{\hat{\beta}}_{11}{\hat{w}}_{o}^{{({1,2})}^{2}}} + {{\hat{\beta}}_{22}{\hat{w}}_{o}^{{({0,3})}^{2}}} - {\hat{w}}_{o}^{{({0,2})}^{4}}} \right)}\end{pmatrix}}}}$

In the following, it will be shown how the aberrations of a spectaclelens are considered in the optimization thereof in a preferable way bythe wavefronts being described in different coordinate systems that arerotated relative to each other. As described with respect to FIG. 1 andFIG. 2, the coordinate systems are preferably defined by theintersection points of the main ray 10 with the refractive surface 14,16, by the refractive surface, and by the direction of the main ray 10.In order to describe an incoming wavefront, the refractive surfaceitself, and the outgoing wavefront for the process of refraction on therefractive surface, preferably three different local Cartesiancoordinate systems (x,y,z), (x,y,z), and (x′,y′,z′) are used. The originof all these coordinate systems preferably coincides with theintersection point of the main ray 10 with the refractive surface. Whilethe systems have the normal direction to the plane of refraction (i.e.the plane in which the incoming and the outgoing main ray are located)as the common axis x=x′=x, the z axis points along the incoming mainray, the z′ axis along the outgoing main ray, and the z axis along thenormal of the refractive surface. The orientations of the axis, axis,and axis are preferably selected such that each system is right-handed(cf. FIG. 5)

At the transition between the coordinate systems, all vector quantitiesv depend on each other via the following relationsv=R(ε) v,v′=R(ε) v   (55)where R designates the rotations about the common x axis and is definedby the three-dimensional rotation matrix

$\begin{matrix}{{R(ɛ)} = \begin{pmatrix}1 & 0 & 0 \\0 & {\cos\; ɛ} & {{- \sin}\; ɛ} \\0 & {\sin\; ɛ} & {\cos\; ɛ}\end{pmatrix}} & (56)\end{matrix}$

In case of a rotation of the coordinate system by the angle α about thez axis, the coordinate transformation is described by

$\begin{matrix}{{\begin{matrix}{\overset{\sim}{x} = {{x\;\cos\;\alpha} - {y\;\sin\;\alpha}}} \\{\overset{\sim}{y} = {{x\;\sin\;\alpha} + {y\;\cos\;\alpha}}}\end{matrix}\mspace{14mu}{or}\mspace{14mu}\begin{pmatrix}\overset{\sim}{x} \\\overset{\sim}{y}\end{pmatrix}} = {{R(\alpha)}\begin{pmatrix}x \\y\end{pmatrix}}} & (57)\end{matrix}$with the rotation matrix

$\begin{matrix}{{R(\alpha)} = \begin{pmatrix}{\cos\;\alpha} & {{- \sin}\;\alpha} \\{\sin\;\alpha} & {\cos\;\alpha}\end{pmatrix}} & (58)\end{matrix}$

Thus, the wavefront {tilde over (w)} in the rotated coordinate system{tilde over (x)}, {tilde over (y)} is described by{tilde over (w)}({tilde over (x)},{tilde over (y)})=w(x({tilde over(x)},{tilde over (y)}),y({tilde over (x)},{tilde over (y)}))  (59)

If one derives the wavefront {tilde over (w)} according to {tilde over(x)},{tilde over (y)}, one obtains the new coefficients ã_(m,k−m)relative to the coefficients a_(m,k−m).

$\begin{matrix}{{\overset{\sim}{a}}_{m,{k - m}} = \left. {\frac{\partial^{k}}{{\partial{\overset{\sim}{x}}^{m}}{\partial{\overset{\sim}{y}}^{k - m}}}{w\left( {{x\left( {\overset{\sim}{x},\overset{\sim}{y}} \right)},{y\left( {\overset{\sim}{x},\overset{\sim}{y}} \right)}} \right)}} \right|_{{\overset{\sim}{x} = 0},{\overset{\sim}{y} = 0}}} & (60)\end{matrix}$

In the second order, the aberrations are preferably represented by thevector

$\begin{matrix}{s = \begin{pmatrix}S_{xx} \\S_{xy} \\S_{yy}\end{pmatrix}} & (61)\end{matrix}$

If the coordinate system is rotated by the angle α, the new aberrations{tilde over (s)} of second order (in the rotated coordinate system({tilde over (x)},{tilde over (y)})) are calculated via

$\begin{matrix}{{\overset{\sim}{s} = {{R_{2}(\alpha)}s}}{with}} & (62) \\{{R_{2}(\alpha)} = \begin{pmatrix}{\cos^{2}\alpha} & {{- 2}\;\cos\;\alpha\;\sin\;\alpha} & {\sin^{2}\alpha} \\{\cos\;\alpha\;\sin\;\alpha} & {{\cos^{2}\alpha} - {\sin^{2}\alpha}} & {{- \cos}\;\alpha\;\sin\;\alpha} \\{\sin^{2}\alpha} & {2\;\cos\;\alpha\;\sin\;\alpha} & {\cos^{2}\alpha}\end{pmatrix}} & (63)\end{matrix}$

For higher orders of the aberrations, the dependency of the newcoefficients a_(m,k−m) on the old coefficients a_(m,k−m) is preferablyexpressed by

$\begin{matrix}{\begin{pmatrix}{\overset{\sim}{a}}_{00} \\{\overset{\sim}{a}}_{01} \\{\overset{\sim}{a}}_{10} \\{\overset{\sim}{a}}_{02} \\{\overset{\sim}{a}}_{11} \\{\overset{\sim}{a}}_{20} \\{\overset{\sim}{a}}_{03} \\{\overset{\sim}{a}}_{12} \\{\overset{\sim}{a}}_{21} \\\vdots\end{pmatrix} = {{R_{Pot}\left( {N,\alpha} \right)}\begin{pmatrix}a_{00} \\a_{01} \\a_{10} \\a_{02} \\a_{11} \\a_{20} \\a_{03} \\a_{12} \\a_{21} \\\vdots\end{pmatrix}}} & (64)\end{matrix}$

The resulting rotation matrix has the block structure, which shows thatthe coefficients a_(m,k−m) of the order k only depend on coefficientsa_(m,k−m) of the same order k. The rotation matrix for the first 15coefficients (N=15) up to the order (k=4) thus reads

$\begin{matrix}{{R_{Pot}\left( {15,\alpha} \right)} = \begin{pmatrix}1 & 0 & \; & \ldots & 0 \\0 & {R_{1}(\alpha)} & \; & \; & \vdots \\\; & \; & {R_{2}(\alpha)} & \; & \; \\\vdots & \; & \; & {R_{3}(\alpha)} & 0 \\0 & \ldots & \; & 0 & {R_{4}(\alpha)}\end{pmatrix}} & (65)\end{matrix}$

The matrix elements of the block structures R_(k)(α) of the first order(k=1) yield the known rotation matrix

$\begin{matrix}{{R_{1}(\alpha)} = {{R(\alpha)} = \begin{pmatrix}{\cos\;\alpha} & {{- \sin}\;\alpha} \\{\sin\;\alpha} & {\cos\;\alpha}\end{pmatrix}}} & (66)\end{matrix}$

In the second order (k=2), the rotation matrix reads

$\begin{matrix}{{R_{2}(\alpha)} = \begin{pmatrix}{\cos^{2}\alpha} & {{- 2}\;\cos\;\alpha\;\sin\;\alpha} & {\sin^{2}\alpha} \\{\cos\;\alpha\;\sin\;\alpha} & {{\cos^{2}\alpha} - {\sin^{2}\alpha}} & {{- \cos}\;\alpha\;\sin\;\alpha} \\{\sin^{2}\alpha} & {2\;\cos\;\alpha\;\sin\;\alpha} & {\cos^{2}\alpha}\end{pmatrix}} & (67)\end{matrix}$in the third order (k=3)

$\begin{matrix}{{R_{3}(\alpha)} = \begin{pmatrix}{\cos^{3}\alpha} & {{- 3}\;\cos^{2}\alpha\;\sin\;\alpha} & {3\;\cos\;\alpha\;\sin^{2}\alpha} & {\sin^{3}\alpha} \\{\cos^{2}\alpha\;\sin\;\alpha} & {{\cos^{3}\alpha} - {2\;\cos\;{\alpha sin}^{2}\alpha}} & {{\sin^{3}\alpha} - {2\;\cos^{2}\alpha\;\sin\;\alpha}} & {\cos\;\alpha\;\sin^{2}\alpha} \\{\cos\;\alpha\;\sin^{2}\alpha} & {- \left( {{\sin^{3}\alpha} - {2\;\cos^{2}\alpha\;\sin\;\alpha}} \right)} & {{\cos^{3}\alpha} - {2\;\cos\;{\alpha sin}^{2}\alpha}} & {\cos^{2}\alpha\;\sin\;\alpha} \\{\sin^{3}\alpha} & {3\;\cos\;\alpha\;\sin^{2}\alpha} & {3\;\cos^{2}\alpha\;\sin\;\alpha} & {\cos^{3}\alpha}\end{pmatrix}} & (68)\end{matrix}$and in the fourth order (k=4)

$\begin{matrix}{{R_{4}(\alpha)} = \begin{pmatrix}{\cos^{4}\alpha} & {{- \cos^{3}}\alpha\;\sin\;\alpha} & {\cos^{2}\alpha\;\sin^{2}\alpha} & {{- \cos}\;\alpha\;\sin^{3}\alpha} & {\sin^{4}\alpha} \\{4\;\cos^{3}\alpha\;\sin\;\alpha} & {{\cos^{4}\alpha} - {3\;\cos^{2}\sin^{2}\alpha}} & \begin{matrix}{2\left( {{\cos\;\alpha\;\sin^{3}\alpha} -} \right.} \\\left. {\cos^{3}\alpha\;\sin\;\alpha} \right)\end{matrix} & \begin{matrix}{- \left( {{\sin^{4}\alpha} -} \right.} \\\left. {3\;\cos^{2}\sin^{2}\alpha} \right)\end{matrix} & {{- 4}\cos\;\alpha\;\sin^{3}\alpha} \\{6\;\cos^{2}\alpha\;\sin^{2}\alpha} & \begin{matrix}{2\left( {{\cos^{3}\alpha\;\sin\;\alpha} -} \right.} \\\left. {\cos\;\alpha\;\sin^{3}\alpha} \right)\end{matrix} & {{\cos^{4}\alpha} - {4\;\cos^{2}\alpha} + {\sin^{4}\alpha}} & \begin{matrix}{3\left( {{\cos\;\alpha\;\sin^{3}\alpha} -} \right.} \\\left. {\cos^{3}\alpha\;\sin\;\alpha} \right)\end{matrix} & {6\;\cos^{2}\alpha\;\sin^{2}\alpha} \\{4\;\cos\;\alpha\;\sin^{3}\alpha} & {- \left( {{\sin^{4}\alpha} - {3\;\cos^{2}\sin^{2}\alpha}} \right)} & \begin{matrix}{{- 2}\left( {{\cos\;\alpha\;\sin^{3}\alpha} -} \right.} \\\left. {\cos^{3}\alpha\;\sin\;\alpha} \right)\end{matrix} & \begin{matrix}{{\cos^{4}\alpha} -} \\{3\;\cos^{2}\sin^{2}\alpha}\end{matrix} & {{- 4}\cos^{3}\alpha\;\sin\;\alpha} \\{\sin^{4}\alpha} & {\cos\;\alpha\;\sin^{3}\alpha} & {\cos^{2}\alpha\;\sin^{2}\alpha} & {\cos^{3}\alpha\;\sin\;\alpha} & {\cos^{4}\alpha}\end{pmatrix}} & (69)\end{matrix}$

The equations (66) to (69) show that the block matrix elementse_(i,j)(α) of the respective rotation matrix R_(k)(α) have the symmetrye_(i,j)(α)=e_(k+2−i,k+2−j)(−α). With c=cos α, s=sin α, the blockmatrices can be simplified to read

$\begin{matrix}{{{R_{1}(\alpha)} = \begin{pmatrix}{c^{k}s^{0}} & * \\{c^{0}s^{k}} & *\end{pmatrix}}{{R_{2}(\alpha)} = \begin{pmatrix}{c^{k}s^{0}} & * & * \\{{kc}^{1}s^{1}} & {c^{2} - s^{2}} & * \\{c^{0}s^{k}} & {c^{1}s^{1}} & *\end{pmatrix}}{{R_{3}(\alpha)} = \begin{pmatrix}{c^{k}s^{0}} & * & * & * \\{{kc}^{k - 1}s^{1}} & {c^{k} - {\left( {k - 1} \right)c^{1}s^{k - 1}}} & * & * \\{{kc}^{1}s^{k - 1}} & {{- s^{k}} + {\left( {k - 1} \right)c^{k - 1}s^{1}}} & * & * \\{c^{0}s^{k}} & {c^{1}s^{k - 1}} & {c^{k - 1}s^{1}} & *\end{pmatrix}}{{R_{4}(\alpha)} = \begin{pmatrix}{c^{k}s^{0}} & * & * & * & * \\{{kc}^{k - 1}s^{1}} & {c^{k} - {\left( {k - 1} \right)c^{2}s^{2}}} & * & * & * \\{2\left( {k - 1} \right)c^{2}s^{2}} & {\left( {k - 1} \right)\left( {{c^{k - 1}s^{1}} - {c^{1}s^{k - 1}}} \right)} & {c^{k} - {4\; c^{2}s^{2}} + s^{k}} & * & * \\{{kc}^{1}s^{k - 1}} & {{- s^{k}} + {\left( {k - 1} \right)c^{2}s^{2}}} & {2\left( {{c^{k - 1}s^{1}} - {c^{1}s^{k - 1}}} \right)} & * & * \\{c^{0}s^{k}} & {c^{1}s^{k - 1}} & {c^{2}s^{2}} & {c^{k - 1}s^{1}} & *\end{pmatrix}}{{R_{5}(\alpha)} = \begin{pmatrix}{c^{k}s^{0}} & * & * & * & * & * \\{{kc}^{k - 1}s^{1}} & {c^{k} - {\left( {k - 1} \right)c^{k - 2}s^{2}}} & * & * & * & * \\{2\;{kc}^{k - 2}s^{2}} & {{\left( {k - 1} \right)c^{k - 1}s^{1}} - {2\left( {k - 2} \right)c^{2}s^{k - 2}}} & {c^{k} - {6\; c^{k - 2}s^{2}} + {3\; c^{1}s^{k - 1}}} & * & * & * \\{2\;{kc}^{2}s^{k - 2}} & \begin{matrix}{{{- \left( {k - 1} \right)}c^{1}s^{k - 1}} +} \\\left. {2\left( {k - 2} \right)c^{k - 2}s^{2}} \right)\end{matrix} & {s^{k} - {6\; c^{2}s^{k - 2}} + {3\; c^{k - 1}s^{1}}} & * & * & * \\{{kc}^{1}s^{k - 1}} & {{- s^{k}} + {\left( {k - 1} \right)c^{2}s^{k - 2}}} & {{{- 2}\; c^{1}s^{k - 1}} + {3\; c^{k - 2}s^{2}}} & {{2\; c^{k - 1}s^{1}} - {3\; c^{2}s^{k - 2}}} & * & * \\{c^{0}s^{k}} & {c^{1}s^{k - 1}} & {c^{2}s^{k - 2}} & {c^{k - 2}s^{2}} & {c^{k - 1}s^{1}} & *\end{pmatrix}}} & (70)\end{matrix}$

In a preferred embodiment, the aberrations are described in the form ofZernike polynomials. In this case, the rotation is performed in thespace of the Zernike polynomials. The wavefront is preferably spanned bythe Zernike polynomials in polar coordinates:

$\begin{matrix}{{{Z_{0,0}\left( {\rho,\varphi} \right)} = 1}{{Z_{1,1}\left( {\rho,\varphi} \right)} = {2\rho\;\cos\;\varphi}}{{Z_{1,{- 1}}\left( {\rho,\varphi} \right)} = {2\rho\;\sin\;\varphi}}{{Z_{2,0}\left( {\rho,\varphi} \right)} = {\sqrt{3}\left( {{2\rho^{2}} - 1} \right)}}{{Z_{2,2}\left( {\rho,\varphi} \right)} = {\sqrt{6}\rho^{2}\cos\; 2\varphi}}{{Z_{2,{- 2}}\left( {\rho,\varphi} \right)} = {\sqrt{6}\rho^{2}\cos\; 2\varphi}}\vdots{with}} & (71) \\{{W\left( {x,y} \right)} = {\sum\limits_{k = 0}^{\infty}\;{\sum\limits_{m}\;{c_{k,m}Z_{k,m}}}}} & (72)\end{matrix}$

The Zernike coefficients corresponding to a wavefront w(x, y) arepreferably determined via the integral

$\begin{matrix}{c_{k}^{m} = {\frac{1}{\pi\; r_{0}^{2}}\underset{pupil}{\int^{\;}\int}{Z_{k}^{m}\left( {\frac{x}{r_{0}},\frac{y}{r_{0}}} \right)}{w\left( {x,y} \right)}{\mathbb{d}x}\ {\mathbb{d}y}}} & (73)\end{matrix}$where r:=√{square root over (x²+y²)}, x=ρ cos φ, y=ρ sin φ, and r₀ thepupil size.

In the preferred representation by means of Zernike polynomials in polarcoordinates, the rotation for the Zernike coefficients is very simple.The vector of Zernike coefficients is transformed by the rotation

$\begin{matrix}{\begin{pmatrix}{\overset{\sim}{a}}_{00} \\{\overset{\sim}{a}}_{01} \\{\overset{\sim}{a}}_{10} \\{\overset{\sim}{a}}_{02} \\{\overset{\sim}{a}}_{11} \\{\overset{\sim}{a}}_{20} \\{\overset{\sim}{a}}_{03} \\{\overset{\sim}{a}}_{12} \\{\overset{\sim}{a}}_{21} \\\vdots\end{pmatrix} = {{R_{Pot}\left( {N,\alpha} \right)}\begin{pmatrix}a_{00} \\a_{01} \\a_{10} \\a_{02} \\a_{11} \\a_{20} \\a_{03} \\a_{12} \\a_{21} \\\vdots\end{pmatrix}}} & (74)\end{matrix}$

In a block matrix representation, the rotation matrix is directly basedon the elementary rotation matrix of equation (57). For N=15, therotation matrix has the form:

For illustration purposes, every block belonging to the same radialorder is framed.

If the wavefront is represented via a series as in equations (70) and(71), a series representation, i.e. a linear combination of thecoefficients a_(m,k−m) results for the integral of equation (72) aswell. If the coefficients c_(k) ^(m) or a_(m,k−m) are summed as vectorsup to a specific order k, a transition matrix T(N) between the Zernikesubspace and the Taylor series subspace of the order k can be indicatedby

$\begin{matrix}{\begin{pmatrix}c_{0,0} \\c_{1,1} \\c_{1,{- 1}} \\c_{2,0} \\c_{2,2} \\c_{2,{- 2}} \\c_{3,1} \\c_{3,{- 1}} \\c_{3,3} \\\vdots\end{pmatrix} = {{{T(N)}\begin{pmatrix}E \\E_{x} \\E_{y} \\E_{xx} \\E_{xy} \\E_{yy} \\E_{xxx} \\E_{xxy} \\\vdots \\E_{{yy}\mspace{11mu}\ldots\mspace{11mu} y}\end{pmatrix}} = {n\;{T(N)}\begin{pmatrix}a_{00} \\a_{01} \\a_{10} \\a_{02} \\a_{11} \\a_{20} \\a_{03} \\a_{12} \\a_{21} \\\vdots\end{pmatrix}}}} & (76)\end{matrix}$with T(N)=Z(N)D(N), where e.g. for N=9

$\begin{matrix}{{D(9)} = \begin{pmatrix}1 & \ldots & \; & \; & \; & \; & \; & \ldots & 0 \\0 & r_{0} & \; & \; & \; & \; & \; & \; & \vdots \\\vdots & \; & r_{0} & \; & \; & \; & \; & \; & \; \\\; & \; & \; & r_{0}^{2} & \; & \; & \; & \; & \; \\\; & \; & \; & \; & r_{0}^{2} & \; & \; & \; & \; \\\; & \; & \; & \; & \; & r_{0}^{2} & \; & \; & \; \\\; & \; & \; & \; & \; & \; & r_{0}^{3} & \; & \; \\\vdots & \; & \; & \; & \; & \; & \; & r_{0}^{3} & \vdots \\0 & \ldots & \; & \; & \; & \; & \ldots & 0 & r_{0}^{3}\end{pmatrix}} & (77)\end{matrix}$designates a matrix that indicates the correct power of the pupilradius. The basic transformation matrix Z(N) is determined by Zernikeexpansion of the power series. Preferably, the following representationis provided for the transformation matrix for N=15:

${Z^{- 1}(15)} = \begin{pmatrix}1 & 0 & 0 & {- \sqrt{3}} & 0 & 0 & 0 & 0 & 0 & 0 & * & 0 & 0 & 0 & 0 \\0 & 2 & 0 & 0 & 0 & 0 & {{- 4}\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & 0 & \vdots & \vdots \\\vdots & 0 & 2 & 0 & 0 & 0 & 0 & {{- 4}\sqrt{2}} & 0 & 0 & 0 & 0 & 0 & \; & \; \\\; & \vdots & 0 & {4\sqrt{3}} & {2\sqrt{6}} & 0 & 0 & 0 & 0 & 0 & * & 0 & * & \; & \; \\\; & \; & \vdots & 0 & 0 & {2\sqrt{6}} & 0 & 0 & 0 & 0 & 0 & * & 0 & \; & \; \\\; & \; & \; & {4\sqrt{3}} & {{- 2}\sqrt{6}} & 0 & 0 & 0 & 0 & 0 & * & 0 & * & \; & \; \\\; & \; & \; & 0 & 0 & 0 & {36\sqrt{2}} & 0 & {12\sqrt{2}} & 0 & 0 & 0 & 0 & \; & \; \\\; & \; & \; & \vdots & \vdots & \vdots & 0 & {12\sqrt{2}} & 0 & {12\sqrt{2}} & 0 & 0 & 0 & \; & \; \\\; & \; & \; & \; & \; & \; & {12\sqrt{2}} & 0 & {{- 12}\sqrt{2}} & 0 & 0 & 0 & 0 & \vdots & \vdots \\\; & \; & \; & \; & \; & \; & 0 & {36\sqrt{2}} & 0 & {{- 12}\sqrt{2}} & 0 & 0 & 0 & 0 & 0 \\\; & \; & \; & \; & \; & \; & \vdots & 0 & \vdots & 0 & * & 0 & * & 0 & * \\\; & \; & \; & \; & \; & \; & \; & \vdots & \; & \vdots & 0 & * & 0 & * & 0 \\\; & \; & \; & \; & \; & \; & \; & \; & \; & \; & * & 0 & * & 0 & * \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & 0 & * & 0 & * & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & * & 0 & * & 0 & *\end{pmatrix}$

In this equation as well, the blocks belonging to the same radial orderare framed for purposes of illustration. It can be seen thatnon-disappearing elements also exist outside the diagonal blocks.However, they do not influence the rotation matrix R_(Pot)(N, α).

In order to determine the rotation matrix R_(Pot)(N, α), R_(Zernike)(N,α) is transformed to the coefficient system of the power seriesdevelopment with equation (76):

$\begin{pmatrix}{\overset{\sim}{a}}_{00} \\{\overset{\sim}{a}}_{01} \\{\overset{\sim}{a}}_{10} \\{\overset{\sim}{a}}_{02} \\{\overset{\sim}{a}}_{11} \\{\overset{\sim}{a}}_{20} \\{\overset{\sim}{a}}_{03} \\{\overset{\sim}{a}}_{12} \\{\overset{\sim}{a}}_{21} \\\vdots\end{pmatrix} = {{{T^{- 1}(N)}\begin{pmatrix}{\overset{\sim}{c}}_{0,0} \\{\overset{\sim}{c}}_{1,1} \\{\overset{\sim}{c}}_{1,{- 1}} \\{\overset{\sim}{c}}_{2,0} \\{\overset{\sim}{c}}_{2,2} \\{\overset{\sim}{c}}_{2,{- 2}} \\{\overset{\sim}{c}}_{3,1} \\{\overset{\sim}{c}}_{3,{- 1}} \\{\overset{\sim}{c}}_{3,3} \\\vdots\end{pmatrix}} = {{{T^{- 1}(N)}{R_{Zernike}\left( {N,\alpha} \right)}\begin{pmatrix}c_{0,0} \\c_{1,1} \\c_{1,{- 1}} \\c_{2,0} \\c_{2,2} \\c_{2,{- 2}} \\c_{3,1} \\c_{3,{- 1}} \\c_{3,3} \\\vdots\end{pmatrix}} = {{T^{- 1}(N)}{R_{Zernike}\left( {N,\alpha} \right)}{T(N)}\begin{pmatrix}a_{00} \\a_{01} \\a_{10} \\a_{02} \\a_{11} \\a_{20} \\a_{03} \\a_{12} \\a_{21} \\\vdots\end{pmatrix}}}}$

From this, it follows thatR _(Pot)(N,α)=T ⁻¹(N)R _(Zernike)(N,α)T(N)  (78)with a block structure of the form

$\begin{matrix}{{R_{Pot}(15)} = \begin{pmatrix}1 & 0 & \; & \ldots & 0 \\0 & {R_{1}(\alpha)} & \; & \; & \vdots \\\; & \; & {R_{2}(\alpha)} & \; & \; \\\vdots & \; & \; & {R_{3}(\alpha)} & 0 \\0 & \ldots & \; & 0 & {R_{4}(\alpha)}\end{pmatrix}} & (79)\end{matrix}$wherein the block matrices are identical with those of equation (70).

FIG. 6 illustrates an exemplary method for individually optimizing aspectacle lens taking higher-order aberrations (HOA) of both the eye andthe spectacle lens into consideration. In a step ST12, not only thelocal aberrations of 2^(nd) order (S′_(xx), S′_(xy), S′_(yy)) but alsothe aberrations of a higher order (K′_(xxx), K′_(xxy), K′_(xyy) etc.) atthe vertex sphere are calculated on the basis of wavefront tracing(ST10).

From these, from the local aberrations, the values for sphere, cylinder,and cylinder axis (sph, zyl, A) of the spectacle lens are calculatedwith the help of Zernike polynomials and/or other suitable metrics,preferably taking the pupil diameter or pupil radius into consideration.Preferably, in a step S14, first of all Zernike coefficients (c₂ ⁰, c₂², c₂ ⁻², . . . ) are determined. Since now also the higher-order localaberrations are known, it is possible to calculate the ideal sph, zyl, Avalues of the spectacle lens for a finite pupil opening, whichpreferably correspond to the above-described transformed values. Boththe connection between the local aberrations (S′_(xx), S′_(xy), S′_(yy),K′_(xxx), K′_(xxy), K′_(xyy), . . . ) and the Zernike coefficients (c₂⁰, c₂ ², c₂ ⁻², . . . ), as it is particularly referred to in step ST14,and the connection between the Zernike coefficients (c₂ ⁰, c₂ ², c₂ ⁻²,. . . ) and the values for sphere (Sph), cylinder (Zyl bzw. Cyl), andcylinder axis (A or α) are provided as functional connections c₂ ⁰, c₂², c₂ ⁻², . . . )=f(r, ′_(xx), S′_(xy), S′_(yy), K′_(xxx), K′_(xxy),K′_(xyy), . . . ) and Sph, Zyl, A=f(r, c₂ ⁰, c₂ ², c₂ ⁻², . . . ) in astep ST18, particularly taking the pupil radius r into consideration.

Now, it is preferred that the pupil size r be specified to be variablefor every visual point. It is particularly preferred that the pupil sizebe specified as a function of the object distance, which in turnrepresents a function of the visual point. This can be based e.g. on thenear reflex, so that with near objects the assumed pupil diameterdecreases.

Preferably, in the refraction determination (ST20), not only the valuesfor sphere, cylinder, and cylinder axis, particularly for distance andnear vision, are determined subjectively, but additionally thehigher-order aberrations (c₂ ⁰, c₂ ², c₂ ⁻², . . . ) are determined withan aberrometer. In a step ST22, the subjective and objective refractiondata are combined particularly considering object distance, direction ofsight, and pupil diameter. Thus, it is possible to calculate ideal(transformed) prescription values (sph, zyl, A) particularly fordifferent pupil diameters depending on the visual point with suitablemetrics. It is particularly preferred that the ideal prescriptions becalculated once and then be deposited as a function of the objectdistance. Moreover, it is preferred that e.g. with the aberrometer alsothe individual pupil diameter be determined under photopic (small pupil)and mesopic (large pupil) conditions. Otherwise, standard values fromliterature have to be used. Subsequently, the spherocylindrical valuesof the spectacle lens (SL) can be combined with those of the eye (ST24)in a known way (combination SL/eye K:K(Ref,Ast)=SL(Sph,Zyl,Axis)−eye(Sph,Zyl,axis). The target function(ST26), in which particularly the target values S(Ref, Ast) provided ina step ST28 are taken into account, preferably remains unchanged. Thedifferences between the combination values K and the target values Sdetermined in step ST24 are particularly taken into account therein:K(Ref,Ast)−S(Ref,Ast).

LIST OF REFERENCE NUMERALS

-   10 main ray-   12 eye-   14 first refractive surface (front surface of the spectacle lens)-   16 second refractive surface (back surface of the spectacle lens)-   18 original wavefront-   20 propagated wavefront-   22 image point-   24 neighboring ray-   ST2, ST4 propagation (and optionally rotation)

The invention claimed is:
 1. A computer-implemented method for designinga spectacle lens and manufacturing the designed spectacle lens,comprising: specifying at least one surface for the spectacle lens to bedesigned; determining the course of a main ray through at least onevisual point (i) of the at least one surface; determining a sphericaland astigmatic aberration outgoing local wavefront set of coefficients(s_(io)) and a higher-order aberration outgoing local wavefront set ofcoefficients (e_(iok)) of the local aberration of a local wavefrontgoing out from the at least one visual point (i) in a surrounding of themain ray; specifying at least one function f(e_(ok)) which assigns ahigher-order aberration propagated local wavefront set of coefficients(e_(pk)) to a higher-order aberration outgoing local wavefront set ofcoefficients (e_(ok)), wherein e_(pk)=B_(k)(e_(ok)+r_(k)), and theproportionality term B_(k) depends on the spherical and astigmaticaberration outgoing local wavefront set of coefficients (s_(io)), butnot on the higher-order aberration outgoing local wavefront set ofcoefficients (e_(ok)), and r_(k) is a remainder term; determining ahigher-order aberration of a local wavefront propagated starting fromthe at least one visual point (i) along the main ray depending on atleast the higher-order aberration outgoing local wavefront set ofcoefficients (e_(iok)) on the basis of the specified function f(e_(ok));designing the at least one surface of the spectacle lens based on thedetermined higher-order aberration of the propagated local wavefront;and manufacturing the designed spectacle lens.
 2. The method accordingto claim 1, wherein determining a spherical and astigmatic aberrationoutgoing local wavefront set of coefficients comprises determining apower vector ${s_{o} = \begin{pmatrix}S_{oxx} \\S_{oxy} \\S_{oyy}\end{pmatrix}},$ wherein determining a higher-order aberration outgoinglocal wavefront set of coefficients comprises determining a coma vector${e_{o\; 3} = \begin{pmatrix}E_{oxxx} \\E_{oxxy} \\E_{oxyy} \\E_{oyyy}\end{pmatrix}},$ and wherein the function$e_{p\; 3} = {{\gamma^{3}\begin{pmatrix}\beta_{y}^{- 3} & {3\beta_{7}^{- 2}\frac{d}{n}S_{xy}} & {3{\beta_{y}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2}} & \left( {\frac{d}{n}S_{xy}} \right)^{3} \\{\beta_{y}^{- 2}\frac{d}{n}S_{xy}} & {\beta_{y}^{- 1}\left( {\frac{1}{\gamma} + {3\left( {\frac{d}{n}S_{xy}} \right)^{2}}} \right)} & {{\frac{2}{\gamma}\frac{d}{n}S_{xy}} + {3\left( {\frac{d}{n}S_{xy}} \right)^{3}}} & {\beta_{x}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2} \\{\beta_{y}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2} & {{\frac{2}{\gamma}\frac{d}{n}S_{xy}} + {3\left( {\frac{d}{n}S_{xy}} \right)^{3}}} & {\beta_{x}^{- 1}\left( {\frac{1}{\gamma} + {3\left( {\frac{d}{n}S_{xy}} \right)^{2}}} \right)} & {\beta_{x}^{- 2}\frac{d}{n}S_{xy}} \\\left( {\frac{d}{n}S_{xy}} \right)^{3} & {3{\beta_{x}^{- 1}\left( {\frac{d}{n}S_{xy}} \right)}^{2}} & {3\beta_{x}^{- 2}\frac{d}{n}S_{xy}} & \beta_{x}^{- 3}\end{pmatrix}}e_{o\; 3}}$${{{with}\mspace{14mu}\gamma} = \frac{1}{\left. {1 - {\frac{d}{n}S_{oxx}} - \left( {\frac{d}{n}S_{oxy}} \right)^{2} - {\frac{d}{n}S_{oyy}} + {\left( \frac{d}{n} \right)^{2}S_{oxx}S_{oyy}}} \right)}},{\beta_{x} = {{\frac{1}{1 - {\frac{d}{n}S_{oxx}}}\mspace{14mu}{and}\mspace{14mu}\beta_{y}} = \frac{1}{1 - {\frac{d}{n}S_{oyy}}}}}$is specified as at least one function f(e_(o3)).
 3. The method accordingto claim 1, wherein determining a spherical and astigmatic aberrationoutgoing local wavefront set of coefficients comprises determining apower vector ${s_{o} = \begin{pmatrix}S_{oxx} \\S_{oxy} \\S_{oyy}\end{pmatrix}},$ wherein determining a higher-order aberration outgoinglocal wavefront set of coefficients comprises determining a coma vector$e_{o\; 3} = \begin{pmatrix}E_{oxxx} \\E_{oxxy} \\E_{oxyy} \\E_{oyyy}\end{pmatrix}$ and determining a spherical aberration vector${e_{o\; 4} = \begin{pmatrix}E_{oxxxx} \\E_{oxxxy} \\E_{oxxyy} \\E_{oxyyy} \\E_{oyyyy}\end{pmatrix}},$ and wherein the function $e_{p\; 4} = {\begin{pmatrix}\beta_{x}^{4} & \ldots & \; & \ldots & 0 \\\vdots & {\beta_{x}^{3}\beta_{y}^{1}} & \; & \; & \vdots \\\; & \; & {\beta_{x}^{2}\beta_{y}^{2}} & \; & \; \\\vdots & \; & \; & {\beta_{x}^{1}\beta_{y}^{3}} & \vdots \\0 & \; & \; & \ldots & \beta_{y}^{4}\end{pmatrix}\left( {e_{o\; 4} + {\frac{d}{n}\begin{pmatrix}{3\left( {{\beta_{x}E_{oxxx}^{2}} + {\beta_{y}E_{oxxy}^{2}} - \frac{S_{oxx}^{4}}{n^{2}}} \right)} \\{3{E_{oxxy}\left( {{\beta_{x}E_{oxxx}} + {\beta_{y}E_{oxyy}}} \right)}} \\\begin{matrix}{{\beta_{x}\left( {{2E_{oxxy}^{2}} + {E_{oxxx}E_{oxyy}}} \right)} + {\beta_{y}\left( {{2E_{oxyy}^{2}} + {E_{oxxy}E_{oyyy}}} \right)} -} \\\left( \frac{S_{oxx}S_{oyy}}{n} \right)^{2}\end{matrix} \\{3{E_{oxyy}\left( {{\beta_{x}E_{oxxy}} + {\beta_{y}E_{oyyy}}} \right)}} \\{3\left( {{\beta_{x}E_{oxyy}^{2}} + {\beta_{y}E_{oyyy}^{2}} - \frac{S_{oyy}^{4}}{n^{2}}} \right)}\end{pmatrix}}} \right)}$$\mspace{20mu}{{{with}\mspace{14mu}\beta_{x}} = {{\frac{1}{1 - {\frac{d}{n}S_{oxx}}}\mspace{14mu}{and}\mspace{14mu}\beta_{y}} = \frac{1}{1 - {\frac{d}{n}S_{oyy}}}}}$is specified as at least one function f(e_(ok)).
 4. The method accordingto claim 1, further comprising determining an angle α between a firstplane of refraction of the main ray at a first surface of the spectaclelens and a second plane of refraction of the main ray at a secondsurface of the spectacle lens, wherein determining a higher-orderaberration comprises: determining a higher-order aberration propagatedlocal wavefront set of coefficients (e_(ipk)) of the local aberration ofthe propagated wavefront; and determining a transformed higher-orderaberration propagated local wavefront set of coefficients ({tilde over(e)}_(ipk)) depending on the determined angle α.
 5. The method accordingto claim 1, which further comprises collecting prescription data V,wherein the prescription data comprises data with respect to sphericalpower Sph_(V), magnitude of the astigmatism Zyl_(V), astigmatism axisAxis_(V), as well as at least one further higher-order refractionHOA_(V).
 6. The method according to claim 5, wherein collectingprescription data comprises collecting first prescription data for afirst object distance and second prescription data for a second objectdistance.
 7. The method according to claim 1, further comprising:specifying an object distance model A1(x, y), where A1 designates theobject distance and (x, y) a visual point or visual spot of thespectacle lens in a specified or specifiable direction of sight;specifying a function r₀=g(A1), which describes the dependence of apupil size r₀ on the object distance A1; determining a pupil size forthe at least one main ray on the basis of the object distance modelA1(x, y) and the specified function r₀=g(A1).
 8. The method according toclaim 1, wherein the spectacle lens is a progressive spectacle lens. 9.A computer program product stored on a non-transitory computer readablemedium and adapted, when loaded and executed on a computer, to perform amethod for designing a spectacle lens according to claim
 1. 10. A methodfor producing a spectacle lens, comprising: designing a spectacle lensaccording to the method for designing a spectacle lens according toclaim 1; and manufacturing the designed spectacle lens.
 11. A device forproducing a spectacle lens, comprising: a calculator configured tocalculate the spectacle lens according to a method for designing aspectacle lens according to claim 1; and the device configured toproduce the spectacle lens.
 12. A device for designing a spectacle lensand manufacturing the designed spectacle lens, comprising: a surfacemodel database configured to specify at least one surface for thespectacle lens to be designed; a main ray determiner configured todetermine the course of a main ray through at least one visual point (i)of said at least one surface; a primary coefficient determinerconfigured to determine a spherical and astigmatic aberration outgoinglocal wavefront set of coefficients (s_(io)) and a higher-orderaberration outgoing local wavefront setoff coefficients (e_(iok)) of thelocal aberration of a local wavefront going out from the at least onevisual point (i) in a surrounding of the main ray; a propagation modeldatabase configured to specify at least one function f(e_(ok)), whichassigns a higher-order aberration propagated local wavefront set ofcoefficients (e_(pk)) to a higher-order aberration outgoing localwavefront set of coefficients (e_(ok)), whereine_(pk)=B_(k)(e_(ok)+r_(k)), and the proportionality term B_(k) dependson the spherical and astigmatic aberration outgoing local wavefront setof coefficients (s_(io)), but not on the higher-order aberrationoutgoing local wavefront set of coefficients (e_(ok)), and r_(k) is aremainder term; a secondary coefficient determiner configured todetermine a higher-order aberration of a local wavefront propagatedstarting from the at least one visual point (i) along the main raydepending on at least the higher-order aberration outgoing localwavefront set of coefficients (e_(iok)) on the basis of the specifiedfunction f(e_(ok)); a calculator or optimizer configured to calculate oroptimize the at least one surface of the spectacle lens based on thedetermined higher-order aberration of the propagated local wavefront;and manufacturing the designed spectacle lens.